Question

For a sequence S_n = 4(7ⁿ – 1), verify whether the sequence is a G.P.

Answer 1

S_n = 4(7ⁿ – 1)

∴ S_n–1 = 4(7^n–1 – 1)

But, t_n = S_n – S_n–1

= 4(7ⁿ – 1) – 4(7^n–1 – 1)

= 4(7ⁿ – 1– 7^{n– 1} + 1)

= 4(7^n–1+1 – 7^n–1)

= 4.7^n–1 (7 – 1)

∴ t_n = 24.7^n–1

∴ t_n+1 = 24(7)^n+1–1

= 24(7)ⁿ

The sequence (t_n) is a G.P., if `("t"_("n" + 1))/"t"_"n"` = constant for all n ∈ N.

∴ `("t"_("n" + 1))/"t"_"n" = (24(7)^"n")/(24(7)^("n" - 1)` 

= 7 = constant, for all n ∈ N

∴ the sequence is a G.P.