Term and sum of a geometric progression

Text

A geometric progression has first term $a_1=5$ and common ratio $q=3$.

• (a) Compute the sixth term $a_6$.
• (b) Compute the sum of the first six terms $S_6$.

Solution

(a) The general term is $a_n=a_1\,q^{\,n-1}$, so $a_6=5\cdot 3^{5}=5\cdot 243=1215.$ (b) The sum of the first $n$ terms is $S_n=a_1\dfrac{q^{n}-1}{q-1}$: $S_6=5\cdot\frac{3^{6}-1}{3-1}=5\cdot\frac{728}{2}=1820.$ $\boxed{\,a_6=1215,\quad S_6=1820\,}$