College Algebra 7th Edition

Published by Brooks Cole
ISBN 10: 1305115546
ISBN 13: 978-1-30511-554-5

Chapter 8, Sequences and Series - Section 8.3 - Geometric Sequences - 8.3 Exercises: 52

Answer

$S_4=0.7488$

Work Step by Step

The given geometric sequence has: $a_2=0.12 \\a_5=0.00096$ Note that using the second term as starting or reference point, the fifth term of the sequence can be computed by multiplying the common ratio $r$ three times to the second term. Thus, $a_5=a_2 \cdot r \cdot r \cdot r \\a_5 = a_2 \cdot r^3$ Substitute the values of $a_2$ and $a_5$ into the equation above to obtain: $a_5 = a_2 \cdot r^3 \\0.00096 = 0.12 \cdot r^3 \\\dfrac{0.00096}{0.12} = \dfrac{0.12r^3}{0.12} \\0.008 = r^3 (0.2)^3=r^3$ Take the cube root of both sides to obtain: $0.2=r$ RECALL: The partial sum $S_n$ (sum of the first $n$ terms) of a geometric sequence is given by the formula: $S_n=a\left(\dfrac{1-r^n}{1-r}\right), r\ne 1$ where $a$ = first term $r$ = common ratio As of now, only the value of $r$ is known. We need to find the value of $a$. Note that the value of the first term an be found by dividing the second term by the common ratio $r$. Thus, $a = \dfrac{a_2}{r}$ Substitute the values of $a_2$ and $r$ to obtain: $a=\dfrac{0.12}{0.2} \\a=0.6$ Now that both $a$ and $r$ are known, the sum of the first 4 terms can be computed using the formula above to obtain: $\require{cancel} S_4 =0.6\left(\dfrac{1-0.2^4}{1-0.2}\right) \\S_4=0.6\left(\dfrac{1-0.0016}{0.8}\right) \\S_4=0.6\left(\dfrac{0.9984}{0.8}\right) \\S_4=0.6\cdot 1.248 \\S_4=0.7488$
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