{"id":2239,"date":"2025-01-02T13:32:12","date_gmt":"2025-01-02T12:32:12","guid":{"rendered":"https:\/\/wiskunst.nl\/?page_id=2239"},"modified":"2025-01-20T16:44:26","modified_gmt":"2025-01-20T15:44:26","slug":"2025-2","status":"publish","type":"page","link":"https:\/\/wiskunst.nl\/index.php\/wiskunde-is-leuk\/nieuwe-artikelen\/artikel-10-1f\/2025-2\/","title":{"rendered":"2025"},"content":{"rendered":"<p><strong><span class=\"collapseomatic \" id=\"id69e0587033814\"  tabindex=\"0\" title=\"Inhoud\"    >Inhoud<\/span><div id=\"target-id69e0587033814\" class=\"collapseomatic_content \"><\/strong><\/p>\n<ul>\n<li><a href=\"#inleiding\">Inleiding\u00a0<\/a><\/li>\n<li><a href=\"#som1\">Som van product tabel<\/a><\/li>\n<li><a href=\"#som2\">Som van derde machten<\/a><\/li>\n<li><a href=\"#som3\">Kwadraat van som<\/a><\/li>\n<li><a href=\"#som4\">Som van derde machten = Kwadraat van som<\/a><\/li>\n<li><a href=\"#som5\">Som der cijfers<\/a><\/li>\n<li><a href=\"#product\">Merkwaardig product<\/a> <\/div><\/li>\n<\/ul>\n<h3><a id=\"inleiding\"><\/a>Inleiding<\/h3>\n<p>2025 is een getal met een aantal interessante eigenschappen. In dit artikel gaan we een aantal van die eigenschappen noemen.<\/p>\n<h3><a id=\"som1\"><\/a>Som van product tabel<\/h3>\n<p>Wanneer je het product tabel van de getallen 1 t\/m 9 maakt en vervolgens al deze producten optelt dan krijg je 2025:<\/p>\n<table style=\"width: 100%;\" width=\"440\">\n<tbody>\n<tr>\n<td style=\"text-align: center; width: 8.44828%; border-style: solid;\" width=\"40\"><span style=\"color: #ff0000;\">\u00d7<\/span><\/td>\n<td style=\"text-align: center; width: 8.44828%; border-style: solid;\" width=\"40\"><strong>1<\/strong><\/td>\n<td style=\"text-align: center; width: 8.44828%; border-style: solid;\" width=\"40\"><strong>2<\/strong><\/td>\n<td style=\"text-align: center; width: 8.62069%; border-style: solid;\" width=\"40\"><strong>3<\/strong><\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\" width=\"40\"><strong>4<\/strong><\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\" width=\"40\"><strong>5<\/strong><\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\" width=\"40\"><strong>6<\/strong><\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\" width=\"40\"><strong>7<\/strong><\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\" width=\"40\"><strong>8<\/strong><\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\" width=\"40\"><strong>9<\/strong><\/td>\n<td style=\"text-align: center; width: 13.1034%; border-style: solid;\" width=\"40\"><span style=\"color: #ff0000;\">S(r)\u00a0<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; width: 8.44828%; border-style: solid;\"><strong>1<\/strong><\/td>\n<td style=\"text-align: center; width: 8.44828%; border-style: solid;\">1<\/td>\n<td style=\"text-align: center; width: 8.44828%; border-style: solid;\">2<\/td>\n<td style=\"text-align: center; width: 8.62069%; border-style: solid;\">3<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">4<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">5<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">6<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">7<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">8<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">9<\/td>\n<td style=\"text-align: center; width: 13.1034%; border-style: solid;\"><span style=\"color: #ff0000;\">45<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; width: 8.44828%; border-style: solid;\"><strong>2<\/strong><\/td>\n<td style=\"text-align: center; width: 8.44828%; border-style: solid;\">2<\/td>\n<td style=\"text-align: center; width: 8.44828%; border-style: solid;\">4<\/td>\n<td style=\"text-align: center; width: 8.62069%; border-style: solid;\">6<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">8<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">10<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">12<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">14<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">16<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">18<\/td>\n<td style=\"text-align: center; width: 13.1034%; border-style: solid;\"><span style=\"color: #ff0000;\">90<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; width: 8.44828%; border-style: solid;\"><strong>3<\/strong><\/td>\n<td style=\"text-align: center; width: 8.44828%; border-style: solid;\">3<\/td>\n<td style=\"text-align: center; width: 8.44828%; border-style: solid;\">6<\/td>\n<td style=\"text-align: center; width: 8.62069%; border-style: solid;\">9<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">12<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">15<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">18<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">21<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">24<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">27<\/td>\n<td style=\"text-align: center; width: 13.1034%; border-style: solid;\"><span style=\"color: #ff0000;\">135<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; width: 8.44828%; border-style: solid;\"><strong>4<\/strong><\/td>\n<td style=\"text-align: center; width: 8.44828%; border-style: solid;\">4<\/td>\n<td style=\"text-align: center; width: 8.44828%; border-style: solid;\">8<\/td>\n<td style=\"text-align: center; width: 8.62069%; border-style: solid;\">12<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">16<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">20<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">24<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">28<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">32<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">36<\/td>\n<td style=\"text-align: center; width: 13.1034%; border-style: solid;\"><span style=\"color: #ff0000;\">180<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; width: 8.44828%; border-style: solid;\"><strong>5<\/strong><\/td>\n<td style=\"text-align: center; width: 8.44828%; border-style: solid;\">5<\/td>\n<td style=\"text-align: center; width: 8.44828%; border-style: solid;\">10<\/td>\n<td style=\"text-align: center; width: 8.62069%; border-style: solid;\">15<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">20<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">25<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">30<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">35<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">40<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">45<\/td>\n<td style=\"text-align: center; width: 13.1034%; border-style: solid;\"><span style=\"color: #ff0000;\">225<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; width: 8.44828%; border-style: solid;\"><strong>6<\/strong><\/td>\n<td style=\"text-align: center; width: 8.44828%; border-style: solid;\">6<\/td>\n<td style=\"text-align: center; width: 8.44828%; border-style: solid;\">12<\/td>\n<td style=\"text-align: center; width: 8.62069%; border-style: solid;\">18<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">24<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">30<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">36<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">42<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">48<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">54<\/td>\n<td style=\"text-align: center; width: 13.1034%; border-style: solid;\"><span style=\"color: #ff0000;\">270<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; width: 8.44828%; border-style: solid;\"><strong>7<\/strong><\/td>\n<td style=\"text-align: center; width: 8.44828%; border-style: solid;\">7<\/td>\n<td style=\"text-align: center; width: 8.44828%; border-style: solid;\">14<\/td>\n<td style=\"text-align: center; width: 8.62069%; border-style: solid;\">21<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">28<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">35<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">42<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">49<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">56<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">63<\/td>\n<td style=\"text-align: center; width: 13.1034%; border-style: solid;\"><span style=\"color: #ff0000;\">315<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; width: 8.44828%; border-style: solid;\"><strong>8<\/strong><\/td>\n<td style=\"text-align: center; width: 8.44828%; border-style: solid;\">8<\/td>\n<td style=\"text-align: center; width: 8.44828%; border-style: solid;\">16<\/td>\n<td style=\"text-align: center; width: 8.62069%; border-style: solid;\">24<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">32<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">40<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">48<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">56<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">64<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">72<\/td>\n<td style=\"text-align: center; width: 13.1034%; border-style: solid;\"><span style=\"color: #ff0000;\">360<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; width: 8.44828%; border-style: solid;\"><strong>9<\/strong><\/td>\n<td style=\"text-align: center; width: 8.44828%; border-style: solid;\">9<\/td>\n<td style=\"text-align: center; width: 8.44828%; border-style: solid;\">18<\/td>\n<td style=\"text-align: center; width: 8.62069%; border-style: solid;\">27<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">36<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">45<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">54<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">63<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">72<\/td>\n<td style=\"text-align: center; width: 8.7931%; border-style: solid;\">81<\/td>\n<td style=\"text-align: center; width: 13.1034%; border-style: solid;\"><span style=\"color: #ff0000;\">405<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; width: 8.44828%;\"><\/td>\n<td style=\"text-align: center; width: 8.44828%;\"><\/td>\n<td style=\"text-align: center; width: 8.44828%;\"><\/td>\n<td style=\"text-align: center; width: 8.62069%;\"><\/td>\n<td style=\"text-align: center; width: 8.7931%;\"><\/td>\n<td style=\"text-align: center; width: 8.7931%;\"><\/td>\n<td style=\"text-align: center; width: 8.7931%;\"><\/td>\n<td style=\"text-align: center; width: 8.7931%;\"><\/td>\n<td style=\"text-align: center; width: 8.7931%;\"><\/td>\n<td style=\"text-align: center; width: 8.7931%;\"><\/td>\n<td style=\"text-align: center; width: 13.1034%;\"><strong><span style=\"color: #ff0000;\">2025<\/span><\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3><a id=\"som2\"><\/a>Som van derde machten<\/h3>\n<p>Stel dat je de volgende kubussen hebt:<\/p>\n<p>een kubus van 1\u00d71\u00d71, van 2\u00d72\u00d72, van 3\u00d73\u00d73, &#8230;, van 8\u00d78\u00d78 en van 9\u00d79\u00d79 en je bouwt daar een piramide van (met de grootste onder t\/m de kleinste boven) dan heb je precies 2025 kubussen van 1\u00d71\u00d71.<\/p>\n<p>Wiskundig ziet dat er zo uit:<\/p>\n<div class=\"wp-katex-eq\" data-display=\"false\">\\sum\\limits_{k=1}^{9}k^{3}=2025<\/div>\n<p><em><span class=\"collapseomatic \" id=\"id69e05870338a5\"  tabindex=\"0\" title=\"Bewijs\"    >Bewijs<\/span><div id=\"target-id69e05870338a5\" class=\"collapseomatic_content \"><\/em><\/p>\n<p>Hier volgt een simpel bewijs voor<\/p>\n<div class=\"wp-katex-eq\" data-display=\"false\">\\sum\\limits_{k=1}^{9}k^{3}=2025<\/div>\n<p>1<sup>3<\/sup>+2<sup>3<\/sup>+3<sup>3<\/sup>+4<sup>3<\/sup>+5<sup>3<\/sup>+6<sup>3<\/sup>+7<sup>3<\/sup>+8<sup>3<\/sup>+9<sup>3<\/sup> = 1+8+27+64+125+216+343+512+729 = 2025<\/p>\n<p>Maar laten we dit maar even veralgemeniseren:<\/p>\n<p>Een algemene formule voor het optellen van derde machten luidt:<\/p>\n<div class=\"wp-katex-eq\" data-display=\"false\">\\sum\\limits_{k=1}^{n}k^{3} = \\frac{n^{2}(n+1)^{2}}{4}= \\frac{n^{2}\\left ( n^{2}+2n+1 \\right )}{4} = \\frac{n^{4}+2n^{3}+n^{2}}{4}<\/div>\n<p>Dit kan bewezen worden door volledige inductie:<\/p>\n<p>a) voor n=1 geldt:<\/p>\n<div class=\"wp-katex-eq\" data-display=\"false\">\\sum\\limits_{k=1}^{1}k^{3} = \\frac{1^{2}(1+1)^{2}}{4} = 1^{3} = \\frac{1\\cdot 4}{4} = \\frac{4}{4} = 1<\/div>\n<p>b) voor n&gt;1 geldt:<\/p>\n<p>I)<\/p>\n<div class=\"wp-katex-eq\" data-display=\"false\">\\sum\\limits_{k=1}^{n+1}k^{3} = \\frac{\\left ( n+1 \\right )^{2}\\left ( n+2 \\right )^{2}}{4} = \\frac{\\left ( n^{2}+2n+1 \\right )\\left ( n^{2}+4n+4 \\right )}{4} = \\frac{n^{4}+4n^{3}+4n^{2}+2n^{3}+8n^{2}+8n}n^{2}+4n+4{4} = \\frac{n^{4}+6n^{3}+13n^{2}+12n+4}{4}<\/div>\n<p>II)<\/p>\n<div class=\"wp-katex-eq\" data-display=\"false\">\\sum\\limits_{k=1}^{n+1}k^{3} = \\sum\\limits_{k=1}^{n}k^{3}+\\left ( n+1 \\right )^{3} = \\frac{n^{4}+2n^{3}+n^{2}}{4}+n^{3}+3n^{2}+3n+1 = \\frac{n^{4}+2n^{3}+n^{2}+4n^{3}+12n^{2}+12n+4}{4} = \\frac{n^{4}+6n^{3}+13n^{2}+12n+4}{4}<\/div>\n<p>En uit de berekeningen blijkt dat I) en II) gelijk zijn.<\/p>\n<p>Uit a) en b) volgt hetgeen we moesten bewijzen ofwel<\/p>\n<p>q.e.d.<\/div>\n<h3><a id=\"som3\"><\/a>Kwadraat van som<\/h3>\n<p>Als je de getallen van 1 t\/m 9 bij elkaar optelt en daar vervolgens het kwadraat van neemt dan krijg je 2025.<\/p>\n<p>In wiskundige notatie:<\/p>\n<div class=\"wp-katex-eq\" data-display=\"false\">\\left(\\sum\\limits_{k=1}^{9}k\\right)^{2}=2025<\/div>\n<p>De som van de opeenvolgende getallen vanaf 1 t\/m 9 = 45.<\/p>\n<p>Ook hier bestaat een algemene formule voor:<\/p>\n<div class=\"wp-katex-eq\" data-display=\"false\">\\sum\\limits_{k=1}^{n}k = \\frac{n\\left ( n+1 \\right )}{2}<\/div>\n<p>Het bewijs voor deze formule vindt u in het artikel over <a href=\"https:\/\/wiskunst.nl\/index.php\/wiskunde-is-leuk\/oude-artikelen\/bewijzen\/\">bewijzen<\/a>.<\/p>\n<h3><a id=\"som4\"><\/a>Som van derde machten = Kwadraat van som<\/h3>\n<p>Zoals we hierboven zagen is<\/p>\n<div class=\"wp-katex-eq\" data-display=\"false\">\\sum\\limits_{k=1}^{9}k^{3} = \\left ( \\sum\\limits_{k=1}^{9}k\\right )^{2}=2025<\/div>\n<p>Is dat nu toeval of niet?<\/p>\n<p>Nee, dat is het niet.<\/p>\n<p>En het bewijs daarvoor is heel simpel:<\/p>\n<p>We weten dat<\/p>\n<div class=\"wp-katex-eq\" data-display=\"false\">\\sum\\limits_{k=1}^{n}k^{3} = \\frac{n^{2}(n+1)^{2}}{4}<\/div>\n<p>en dat<\/p>\n<div class=\"wp-katex-eq\" data-display=\"false\">\\sum\\limits_{k=1}^{n}k = \\frac{n\\left ( n+1 \\right )}{2}<\/div>\n<p>Welnu, dan is<\/p>\n<div class=\"wp-katex-eq\" data-display=\"false\">\\left ( \\sum\\limits_{k=1}^{n}k \\right )^{2} = \\left ( \\frac{n\\left ( n+1 \\right )}{2} \\right )^{2} = \\frac{n^{2}\\left ( n+1 \\right )^{2}}{2^{2}} = \\frac{n^{2}\\left ( n+1 \\right )^{2}}{4} = \\sum\\limits_{k=1}^{n}k^{3}<\/div>\n<h3><a id=\"som5\"><\/a>Som der cijfers<\/h3>\n<p>We hebben hierboven gezien dat we in de formules voor <em>n<\/em> steeds 9 moeten invullen om tot 2025 te komen.<\/p>\n<p>Ook de product tabel loopt tot en met 9.<\/p>\n<p>En wat wil nu het geval?<\/p>\n<p>De som der cijfers van 2025 = 2+0+2+5 = 9.<\/p>\n<h3><a id=\"product\"><\/a>Merkwaardig product<\/h3>\n<p>Het getal 2025 is ook te schrijven als het volgende merkwaardige merkwaardige product:<\/p>\n<p>(20+25)<sup>2<\/sup> = 45<sup>2<\/sup> = 2025<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Inleiding 2025 is een getal met een aantal interessante eigenschappen. In dit artikel gaan we een aantal van die eigenschappen noemen. Som van product tabel Wanneer je het product tabel van de getallen 1 t\/m 9 maakt en vervolgens al deze producten optelt dan krijg je 2025: \u00d7 1 2 3 4 5 6 7 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":2065,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"templates\/template-full-width.php","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"class_list":["post-2239","page","type-page","status-publish","hentry"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>2025 - Wiskunst<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/wiskunst.nl\/index.php\/wiskunde-is-leuk\/nieuwe-artikelen\/artikel-10-1f\/2025-2\/\" \/>\n<meta property=\"og:locale\" content=\"nl_NL\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"2025 - Wiskunst\" \/>\n<meta property=\"og:description\" content=\"Inleiding 2025 is een getal met een aantal interessante eigenschappen. 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