Question

The formula for the nth term b of a geometric series is b=arn-1. Find n when b=1,024, a=16 and r=2.

Answer 1

`\bf b=ar^(n-1)~~ \begin{cases} b=1024\\ a=16\\ r=2 \end{cases}\implies 1024=16(2^(n-1))~~ \begin{cases} 1024=&2^(10)\\ 16=&2^4 \end{cases} \\\\\\ 2^(10)=2^4(2^(n-1))\implies \cfrac{2^(10)}{2^4}=2^(n-1)\implies 2^(10)\cdot 2^(-4)=2^(n-1)\implies 2^(10-4)=2^(n-1) \\\\\\ 2^6=2^(n-1)\implies \stackrel{\textit{same base, the exponents must be the same}}{6=n-1\implies 7=n}`