Question

1+3+5+7 = (___)^2


solve it​

Answer 1

To solve -

Find the sum of the following series -

=> 1 + 3 + 5 + 7 ....... a_n

Solution -

Given series :

=> 1 + 3 + 5 + 7 ....... a_n

for a_k = 2k + 1 iff 1 ≤ k ≤ n .

This series is an ap .

Here , the initial term , a is 1.

The required common difference is 2 .

Now, we know that the sum of an ap can be expressed as -

S_n = [ n / 2 ] { 2a + ( n - 1 ) d }

Substituting the given values of a and d,

=> S_n = [ n / 2 ] { 2 + 2n - 2 }

=> S_n = [ n / 2 ] { 2n }

=> S_n = n²

Thus , the sum of the sequence - 1 + 3 + 5 + 7 + ..... is n² where n is the last term.

This is the required answer .

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Additional Information -

1 + 2 + 3 + ....... + n = n ( n + 1 ) / 2.

1² + 2² + 3² + ....... + n² = n( n + 1 )( 2n + 1 ) / 6

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