Q, 36 - Calculus

Question

In Exercises 31–36 provide the first five terms of the given sequence. Unless specified, assume that the first term has index $1$. 

$\left\{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots +\frac{1}{2^k}\right\}$

Step-by-Step Solution

Answer

The first five terms are $\frac{1}{2},\frac{1}{4},\frac{1}{8},\frac{1}{16},\frac{1}{32}.$

Step 1. Given information.

The given sequence is $\left\{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots +\frac{1}{2^k}\right\}$.

Step 2. First term.

The general term of the sequence is $a_k=\frac{1}{2^k}.$

For first term, $k=1,$

$a_1=\frac{1}{2^1}$

Step 3. Remaining terms.

Now, when,

$$\begin{gathered} k=2, \\ {a}_2=\frac{1}{2^2}=\frac{1}{4 \\ } \\ k=3, \\ {a}_3=\frac{1}{2^3}=\frac{1}{8} \\ k=4, \\ {a}_4=\frac{1}{2^4}=\frac{1}{16} \\ k=5, \\ {a}_5=\frac{1}{2^5}=\frac{1}{32} \end{gathered}$$