Question

the sum of first six terms of an AP is 42. The ratio of its 10th and its 30th term is 1:3. Calculate the first and the 13th term of the AP

Thj

Answer 1

Explanation:

EXPLANATION.

Sum of first 6 terms of an Ap = 42.

$\sf : \implies \: formula \: of \: sum \: of \: n \: terms \: of \: an \: ap \\ \\ \sf : \implies \: s_{n} \: = \frac{n}{2} (2a + (n - 1)d) \\ \\ \sf : \implies \: \: s_{6} \: = \frac{6}{2}(2a \: + 5d) = 42 \\ \\ \sf : \implies \: 3(2a + 5d) = 42\\ \\ \sf : \implies \: 6a \: + 15d \: = 42 \: \: \: \\ \\ \sf : \implies \: 2a \: + 5d \: = 14 \: \: \: .....(1)$

The ratio of it's 10th and 30th term is 1:3.

$\sf : \implies \: \dfrac{ t_{10} }{ t_{30}} = \dfrac{1}{3} \\ \\ \sf : \implies \: \frac{a + 9d}{a + 29d} = \frac{1}{3} \\ \\ \sf : \implies \: 3a \: + 27d \: = a \: + 29d \\ \\ \sf : \implies \: 2a \: = 2d \\ \\ \sf : \implies \: a \: = d \: \: \: .....(2)$

$\sf : \implies \: from \: equation \: (1) \: \: \: and \: \: \: (2) \: \: \: we \: get \\ \\ \sf : \implies \: 2a \: + \: 5a \: = 14 \\ \\ \sf : \implies \: 7a \: = 14 \\ \\ \sf : \implies \: a \: = 2 \\ \\ \sf : \implies \: d \: = 2$

=> 13th term of an Ap

=> a + 12d

=> 2 + 12 X 2

=> 26

Therefore,

First term = 2

13 th term = 26.