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Big Ideas Math Algebra 2, 2014 View details

3. Analyzing Geometric Sequences and Series

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Exercise 60 Page 431

a Let

a_{n}

be the number of divers in the

n^{th}

ring. To determine a rule for the given geometric sequence, we will use the formula for the explicit rule of geometric sequences.

a_{n} = a_{1} r^{n - 1}

In our case, since there are

5

divers in the first ring the first term is

a_{1} = 5 .

Because each ring after the first has twice as many skydivers as the preceding ring, its common ratio is

r = 2 .

Next, we will substitute values for

a_{1}

and

r

into the explicit rule of geometric sequences.

a_{n} = a_{1} r^{n - 1}

SubstituteValues

Substitute values

a_{n} = 5 (2)^{n - 1}

b We are asked to find the total number of skydivers when there are four rings. From Part A we know that the number of skydivers in the

n^{th}

ring is a geometric sequence ,with

a_{1} = 5

and

r = 2 .

Next, we will use the formula for finding partial sums of geometric series.

S_{n} = \frac{a _{1} ( 1 - r ^{n} )}{1 - r}

We want to calculate the sum of the first

4

terms. So,

n = 4 .

Let's substitute the values and find the sum.

S_{n} = \frac{a _{1} ( 1 - r ^{n} )}{1 - r}

SubstituteValues

Substitute values

S_{4} = \frac{5 ( 1 - 2 ^{4} )}{1 - 2}

▼

Simplify right-hand side

CalcPow

Calculate power

S_{4} = \frac{5 ( 1 - 1 6 )}{1 - 2}

SubTerms

Subtract terms

S_{4} = \frac{5 ( - 1 5 )}{- 1}

Multiply

S_{4} = \frac{- 7 5}{- 1}

ReduceFrac

$\frac{a}{b} = \frac{a / ( - 1 )}{b / ( - 1 )}$

S_{4} = \frac{7 5}{1}

DivByOne

$\frac{a}{1} = a$

S_{4} = 75

Therefore, there are

75

skydivers in all four rings.

Subchapter links

Communicate Your Answer p.425

Monitoring Progress p.426-429

Exercises p.430-432