Find the sum of first 25 terms of the A.P.

Find the sum of first 25 terms of the A.P. whose nth term is given by an = 5 + 6n. Also, find the ratio of 20th term to 45th term. 

Answer

a_n= 5 + 6n

n = 1, a₁ (1^st term) = 5 + 6(1) = 11

n = 2, a₂ (2^nd term) = 5 + 6(2) = 17

⇒ d = a₂ – a₁ = 17 – 11 = 6

$$ S_n = \frac{n}{2} [2a + (n – 1)d] $$

$$ S_{25} = \frac{25}{2} [2(11) + (25 – 1)6] $$

$$ = \frac{25}{2} [22 + 144] $$

S₂₅ = 2075

$$ \frac{a_{20}}{a_{45}} = \frac{5+6(20)}{5+6(44)} = \frac{125}{275} = \frac{5}{11} $$

∴ The required ratio is 5:11