Question

The third and sixth term of an exponential sequence are 1/4 and 1/3 respectively. Find the first term and the common ratio. Find the  eighth term in the sequence.

Answer 1

An interesting geometric progression problem that involves some algebra.

- PeterPJF

Answer 2

ar² = \(\frac{1}{4}\)

ar⁵ = \(\frac{1}{3 }\)

\(\frac{ar^5}{ar^2}\) = \(\frac{4}{3 }\)

r³ = \(\frac{4}{3 }\)  

∴ a = \(\frac{1}{4}\)r^-2 = \(\frac{1}{4}(\frac{3}{4})^\frac{2}{3}\)

T₈ = ar⁷ = ar²(r⁵) = \(\frac{1}{4}(\frac{4}{3})^\frac{5}{3}\)

= \(\frac{1}{4}\) x \(\frac{4}{3}(\frac{4}{3})^\frac{2}{3}\)= \(\frac{1}{3}(\frac{4}{3})^\frac{2}{3}\)