Q) Find the sum to 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.


A) 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) ]
..
..
.. solving further..

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

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