Question

how to solve this question plzz tell me in detail

Answer 1

Given Pattern :

$\mathsf{\bigstar\;\;1^2 = \dfrac{1}{6}[1 \times (1 + 1) \times (2 \times 1 + 1)]}$

$\mathsf{\bigstar\;\;1^2 + 2^2 = \dfrac{1}{6}[2 \times (2 + 1) \times (2 \times 2 + 1)]}$

$\mathsf{\bigstar\;\;1^2 + 2^2 + 3^2 = \dfrac{1}{6}[3 \times (3 + 1) \times (2 \times 3 + 1)]}$

$\mathsf{\bigstar\;\;1^2 + 2^2 + 3^2 + 4^2 = \dfrac{1}{6}[4 \times (4 + 1) \times (2 \times 4 + 1)]}$

Based on the above pattern, We can write general formula as :

$\mathsf{\bigstar\;\;1^2 + 2^2 + 3^2 + . . . + n^2 = \dfrac{1}{6}[n \times (n + 1) \times (2 \times n + 1)]}\\\\\mathsf{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(where\;n\;\epsilon\;N)}$

Question : Find the value of  1² + 2² + 3² + 4² + . . . . . . + 10²

We can notice that : Value of n = 10 for above question

According to the pattern, The given question can be written as :

$\mathsf{\implies \dfrac{1}{6}[10 \times (10 + 1) \times (2 \times 10 + 1)]}$

$\mathsf{\implies \dfrac{1}{6}[10 \times (11) \times (20 + 1)]}$

$\mathsf{\implies \dfrac{1}{6}[110 \times 21]}$

$\mathsf{\implies \dfrac{1}{2}[110 \times 7]}$

$\mathsf{\implies 55 \times 7}$

$\mathsf{\implies 385}$

Question : Find the value of  5² + 6² + 7² + 8² + 9² + 10² + 11² + 12²

The Above question can be written as :

★  [1² + 2² + 3² + 4² + . . . . . + 12²] - [1² + 2² + 3² + 4²]

According to the Pattern :

$\mathsf{\bigstar\;\;1^2 + 2^2 + 3^2 +.\;.\;.\;.+12^2 = \dfrac{1}{6}[12 \times (12 + 1) \times (2 \times 12 + 1)]}\\\\\mathsf{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(n = 12)}$

$\mathsf{\bigstar\;\;1^2 + 2^2 + 3^2 + 4^2 = \dfrac{1}{6}[4 \times (4 + 1) \times (2 \times 4 + 1)]}\\\\\mathsf{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(n = 4)}$

$\mathsf{\implies \dfrac{1}{6}[12 \times (12 + 1) \times (2 \times 12 + 1)] - \dfrac{1}{6}[4 \times (4+ 1) \times (2 \times 4 + 1)]}$

$\mathsf{\implies \dfrac{1}{6}[12 \times 13 \times (24 + 1)] - \dfrac{1}{6}[4 \times 5 \times (8 + 1)]}$

$\mathsf{\implies \dfrac{1}{6}[12 \times 13 \times 25] - \dfrac{1}{6}[4 \times 5 \times 9]}$

$\mathsf{\implies [2 \times 13 \times 25] - \dfrac{1}{2}[4 \times 5 \times 3]}$

$\mathsf{\implies [26 \times 25] - [2 \times 5 \times 3]}$

$\mathsf{\implies 650 - 30}$

$\mathsf{\implies 620}$