Question

the sum of 20 terms of a series of which every even term is 2 times the term before it and every odd term is 3 times the term before it is, the first term being unity is what?​

Answer 1

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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