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Sequences and Series

Multiple Choice Questions

381.

Let S_n denotes the sum of first n terms of an AP.If S₄ = - 34, S₅ = - 60 and S₆ = - 93, then the common difference and the first term of the AP are respectively

- 7, 2
7, - 4
7, - 2
- 7, - 2

382.
An AP has the property that the sum of first ten terms is half the sum of next ten terms. If the second term is 13, then the common difference is

3

2

5

4

Let a and d be the first term and common difference, respectively.

Given, T₂ = 13 $\Rightarrow$ a + d = 13 ...(i)

According to the question,

$(a_{1} + a_{2} . . . + a_{10}) = \frac{1}{2} (a_{11} + a_{12} + a_{20}) \Rightarrow 2 (a_{1} + a_{2} + . . . + a_{10}) = a_{11} + a_{12} + a_{20} \Rightarrow 3 (a_{1} + a_{2} + . . . + a_{10}) = (a_{1} + a_{2} + . . . + a_{20}) (add a_{1} + a_{2} + . . . + a_{10} on both sides) \Rightarrow 3 \times \frac{10}{2} [2 a + 9 d] = \frac{20}{2} [2 a + 19] d \Rightarrow 6 a + 27 d = 4 a + 38 d \Rightarrow 2 a = 11 d . . . (ii) From Eqs . (i) and (ii), we get \frac{11 d}{2} + d = 13 \Rightarrow d = 2$