Quadratic And Sequence Linkage

Question

The sum of the first n terms of a certain sequence equals S_n = 3n^2 + 5n for every positive integer n; what is its tenth term?

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When the sum of the first n terms is known as a function of n, the nth term is recovered by the difference T_n = S_n - S_{n-1}, a key JEE Advanced bridge between partial sums and terms. Here S_n = 3n^2 + 5n, so S_10 = 300 + 50 = 350 and S_9 = 243 + 45 = 288. The tenth term is T_10 = 350 - 288 = 62. Option 65 wrongly evaluates S_10 - S_8. Option 59 results from a sign slip in S_9. Option 68 mistakenly adds rather than subtracts. Hence T_10 = 62. The quadratic form of S_n also reveals the sequence is arithmetic, since a quadratic partial-sum corresponds to a linear term. More generally, the degree of the general term is always one less than the degree of the partial-sum polynomial, which is a discrete analogue of differentiation lowering the degree of a polynomial, a parallel JEE Advanced likes to surface. Plausibility check: the general term is T_n = S_n - S_{n-1} = 6n + 2, giving T_10 = 62, and this linear formula confirms a constant common difference of 6, consistent with an arithmetic progression.

Quadratic And Sequence Linkage

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