Find the sum of 2n terms of the series whose every even term is ‘a’ times the term before it and every odd term is ‘c’ times the term before it, the first term being unity.

Question

Philoid · Accepted Answer

Let T indicate a term of the progression.
T_1, T_2, T_3, ..., T_n, ...T_2n
T₁ = 1
T₂ = a
T₃ = ca
T₄ = c.a²
T₅ = c².a²
T_k if k is even =

T_{2n =}

S_2n = 1 + a + ca + c.a² + c².a² + c².a³ .....aⁿ. c^{n - 1}
= 1 + [ a + c.a² + c².a³.... + aⁿ.c^{n - 1} ] + [ ca + c².a² + c³.a³..... + c^{n - 1}.a^{n - 1}]
The sum of a G.P. =

For a + c.a² + c².a³.... + aⁿ.c^{n - 1}

a = a, r = ca, n = n

⇒

For [ ca + c².a² + c³.a³..... + c^{n - 1}.a^{n - 1}]

a = ca, r = ca, n = n

⇒

∴ The required result

⇒