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Sum of certain consecutive odd positive integers is 57^{2}-13

Question

Sum of certain consecutive odd positive integers is $5 7^{2} - 1 3^{2}$ . Find the first integer of that series .

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Sum of certain consecutive odd positive integers is

5 7^{2} - 1 3^{2}

. Find the first integer of that series .

14 mins ago

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14 mins ago

Text solutionVerified

Let the odd integers be

2 m + 1, 2 m + 3, 2 m + 5

, ... and let their number be

n

. Then

5 7^{2} - 1 3^{2} = [n /2] [2 (2 m + 1) + (n - 1) 2] = n (2 m + n) = 2 mn + n^{2} \Rightarrow 5 7^{2} - 1 3^{2} = [n + m]^{2} - m^{2}

Hence ,

m = 13

and

n + m = 57

n = 57 - 13 = 44

Therefore , the required odd integers are

27, 29, 31, ..., 113

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Question Text	Sum of certain consecutive odd positive integers is $5 7^{2} - 1 3^{2}$ . Find the first integer of that series .
Answer Type	Text solution:1
Upvotes	150