For a sequence, if tn = 5^n-2/7^n-3, verify whether the sequence is a G.P. If it is a G.P., find its first term and the common ratio.

Question

For a sequence, if \(t_n=\frac{5^{n-2}}{7^{n-3}}\), verify whether the sequence is a G.P. If it is a G.P., find its first term and the common ratio.

Answer 1

The sequence (t_n ) is a G.P.

if \(\frac{t_{n+1}}{t_n}\) = constant for all n ∈ N.

Now, \(t_n=\frac{5^{n-2}}{7^{n-3}}\) 

 = 5/7 = constant, for all n ∈ N.

\(\therefore\) the sequence is a G.P. with common ration = 5/7

\(\therefore\) first term = t₁ = \(\frac{5^{1-2}}{7^{1-3}}=\frac{5^{-1}}{7^{-2}}=\frac{7^2}5=\frac{49}5\)