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nth Term of A.P

If the third ...

Question

If the third term of an A.P is $5$ and the ratio of its $6 t h$ term to the $10 t h$ term is $7 : 13$ then find the sum of first $20$ terms of his A.P.

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Solution

$T_{3} = a + 2 d = 5 - (i) T_{6} : T_{10} = (a + 5 d) : (a + 9 d) = 7 : 13 \Rightarrow \frac{a + 5 d}{a + 9 d} = \frac{7}{13} \Rightarrow 13 a + 65 d = 7 a + 63 d \Rightarrow 6 a = - 2 d \Rightarrow - 3 a = d - (i i)$
Putting $(i i)$ in $(i)$ we have
$a + 2 d = 5 \Rightarrow a - 6 a = 5 a = - 1 ∴ d = 3 S_{20} = \frac{20}{2} (2 a + (20 - 1) d) = 10 (- 2 + 19 \times 3) = 10 \times 55 = 550$

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