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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 12: Q 12.51 (page 1191)

Find the sum.
${鈭� 6}_{i = 1}^{15} {(2)}^{i}$

Short Answer

Expert verified

Sum is equal to $\begin{array}{rcl} 393204 \end{array}$

Step by step solution

01

Step 1. Find the first term and the common ratio.

Consider the sequence ${鈭� 6}_{i = 1}^{15} {(2)}^{i}$

Let role="math" localid="1645297852087" $a_{1}, r$ denote the first term and the common ratio respectively.

$\begin{array}{rcl} a_{1} & = & 6 {(2)}^{1} \\ = & 12 \\ a_{2} & = & 6 {(2)}^{2} \\ = & 24 \end{array}$ So,

$\begin{array}{rcl} r & = & \frac{24}{12} \\ = & 2 \end{array}$

02

Step 2. Find the sum.

Consider the sequence ${鈭�}_{i = 1}^{15} 6 {(2)}^{i}$ .

Sum of the first $n$ terms is given by $S_{n} = \frac{a_{1} (1 - r^{n})}{1 - r}$

So,

$\begin{array}{rcl} {鈭�}_{i = 1}^{15} 6 {(2)}^{i} & = & \frac{12 (1 - 2^{15})}{1 - 2} \\ = & - 12 (1 - 32768) \\ = & 393204 \end{array}$

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