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Question 5 :Let t_{1}, t_{2},.....be real numbers such that t_{1}+ t_{2} +... + t_{n} = 2n^{2} + 9n + 13, for every positive integer n ≥ 2. If t_{k}=103, then k equals[TITA]

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Let t_{1}, t_{2},.....be real numbers such that t_{1}+ t_{2} +... + t_{n} = 2n^{2} + 9n + 13

If t_{k} = 103 we have to find the value of k.

Let us assume this as sum to n terms i.e.

S_{n} = t_{1}+ t_{2} +... + t_{n} = 2n^{2} + 9n + 13

Given an expression in terms of n we can also take sum to n-1 terms i.e.

S_{n-1} = t_{1}+ t_{2} +... + t_{n-1} = 2(n-1)^{2} + 9(n-1) +13

This S_{n} and S_{n-1} can be subtracted one from the another such that,

S_{n} = t_{1}+ t_{2} +... + t_{n} = 2n^{2} + 9n + 13

S_{n-1} = t_{1}+ t_{2} +... + t_{n-1} = 2(n-1)^{2} + 9(n-1) +13

S_{n} - S_{n-1} = t_{n} = 2n^{2} + 9n + 13 - 2(n-1)^{2} - 9(n-1) - 13

S_{n} - S_{n-1} = t_{n} = 2n^{2} + 9n + 13 - 2(n-1)^{2} - 9n + 9 - 13

S_{n} - S_{n-1} = t_{n} = 2[n^{2} - (n-1)^{2}] + 9

S_{n} - S_{n-1} = t_{n} = 2[n^{2} - n^{2} + 2n - 1] + 9

t_{n} = 2[2n - 1] + 9

t_{n} = 4n - 2 + 9

t_{n} = 4n + 7

So t_{k} = 4k + 7 = 103

k = \\frac{103 - 7}{4})

k = 24

The 24^{th} term is 103

The question is **"Let t _{1}, t_{2},.....be real numbers such that t_{1}+ t_{2} +... + t_{n} = 2n^{2} + 9n + 13, for every positive integer n ≥ 2. If t_{k}=103, then k equals [TITA] "**

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