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Big Ideas Math Algebra 2, 2014 View details

3. Analyzing Geometric Sequences and Series

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Exercise 4 Page 425

Practice makes perfect

a We are asked to find the sum of the following finite series of a geometric sequence.

S = 1 + 2 + 4 + 8 + \dots + 4096 + 8192

Since each consecutive term is

2

times larger than the previous one, the common ratio of the sequence is

r = 2 .

We learned that we should multiply both sides of the equation by the common ratio and then calculate

S - r S .

In our case

S - r S

S - 2 S .

Let's do it!

S = 1 + 2 + 4 + 8 + \dots + 4096 + 8192

MultEqn

$LHS \cdot 2 = RHS \cdot 2$

2 S = 2 + 4 + 8 + \dots + 8192 + 16384

Now, let's write

S - 2 S .

S - 2 S = (1 + 2 + 4 + 8 + \dots + 4096 + 8192) - (2 + 4 + 8 + \dots + 8192 + 16384)

Notice that almost every term on the right-hand side is canceled.

S - 2 S = 1 - 16384

Finally, we will find

S!

S - 2 S = 1 - 16384

SubTerms

Subtract terms

- S = - 16383

MultEqn

$LHS \cdot (- 1) = RHS \cdot (- 1)$

S = 16383

The sum is equal to

16383 .

b We are asked to find the sum of the following finite series of a geometric sequence.

S = 0.1 + 0.01 + \dots + 1 0^{- 9} + 1 0^{- 10}

Since each consecutive term is

10

times smaller than the previous one, the common ratio of the sequence is

r = \frac{1}{1 0} = 0.1 .

We learned that we should multiply both sides of the equation by the common ratio and then calculate

S - r S .

In our case

S - r S

S - 0.1 S .

Let's do it.

S = 0.1 + 0.01 + \dots + 1 0^{- 9} + 1 0^{- 10}

MultEqn

$LHS \cdot 0.1 = RHS \cdot 0.1$

0.1 S = 0.01 + \dots + 1 0^{- 10} + 1 0^{- 11}

Now, let's write

S - 0.1 S .

S - 0.1 S = (0.1 + 0.01 + \dots + 1 0^{- 9} + 1 0^{- 10}) - (0.01 + \dots + 1 0^{- 10} + 1 0^{- 11})

Notice that almost every term on the right-hand side is canceled.

S - 0.1 S = 0.1 - 1 0^{- 11}

Finally, we will find

S!

S - 0.1 S = 0.1 - 1 0^{- 11}

▼

Solve for

S

SubTerms

Subtract terms

0.9 S = 0.1 - 1 0^{- 11}

MultEqn

$LHS \cdot 10 = RHS \cdot 10$

9 S = 1 - 10 \cdot 1 0^{- 11}

$a = a^{1}$

9 S = 1 - 1 0^{1} \cdot 1 0^{- 11}

MultPow

$a^{m} \cdot a^{n} = a^{m + n}$

9 S = 1 - 1 0^{1 - 11}

SubTerms

Subtract terms

9 S = 1 - 1 0^{- 10}

DivEqn

$LHS / 9 = RHS / 9$

S = \frac{1 - 1 0 ^{- 10}}{9}

The sum is equal to

\frac{1 - 1 0 ^{- 10}}{9} .

Subchapter links

Communicate Your Answer p.425

Monitoring Progress p.426-429

Exercises p.430-432

Exercise 4 Page 425

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