Question

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

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Answer 1

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Answer 2

Let T indicate a term of the progression.

T1 T2 T3.......Tn......T2n

T1 = 1

T2 = a

T3 = ca

T4 = c.a^2

T5 = c^2.a^2

Tk if k is even = a^(k/2). c^(k/2 - 1)

T2n = a^(2n/2).c^(2n/2 -1)

T2n = a^n. c^(n-1)

S 2n = 1 + a + ca + c.a^2 + c^2.a^2 + c^2.a^3 .....a^n. c^(n-1)

= 1 + [ a + ca^2 + c^2.a^3 ....+ a^n.c^(n-1) ] + [ ca + c^2.a^2 + c^3.a^3..... + a^(n-1). c^(n-1) ]

= 1 + [ a.(a^n.c^n - 1) / (ac - 1) ] + [ ac( a^(n-1).c^(n-1) - 1) / (ac - 1) ]

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.. solving further..

---> S 2n = (a^n.c^n - 1)(a + 1) / (ac - 1)