1^2 +2^2+••••+n^2=1/6n(n+1)(2n+1) using maths induction ​

Question

1^2 +2^2+••••+n^2=1/6n(n+1)(2n+1)
using maths induction ​

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Thạch Thảo 5 years 2021-08-02T03:16:52+00:00 1 Answers 16 views 0

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  1. Sapo1
    0
    2021-08-02T03:18:26+00:00
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    Hello,

    if\ n=1\ then\ 1^2=1\ and\ \dfrac{1}{6}*1*2*3=1:\ true\ for\ n=1\\

    We suppose the property true for n:

    1²+2²+…+n²=n(n+1)(2n+1) / 6

    and we are going to demonstrate that the property is true for n+1:

    1²+2²+..+(n+1)²=(n+1)*(n+2)*(2n+3)/6

    1^2+2^2+...+n^2+(n+1)^2\\\\=n*(n+1)*(2n+1)/6+(n+1)^2\\\\=(n+1)/6*[n(2n+1)+6n+6]\\\\=(n+1)/6*(2n^2+7n+6)\\\\=(n+1)(n+2)(2n+3)/6\\

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