Exercise 16 Page 915

Practice makes perfect

Let a_n be Kyla's salary after n years. Because her salary increases by a percentage each year, the sequence a_n is a geometric sequence. Let's identify the two key features of geometric sequences for the given situation.

The common ratio r will be the rate at which her salary grows.
The first term a_1 will be the value of her salary after one year.

Her yearly raise is 6 %, so her salary grows by 1+0.06=1.06 times each year. If we multiply her current salary by this rate, we can find her salary after one year. 50 000 * 1.06 = 53 000 Therefore, the common ratio of a_n is r= 1.06 and the first term is a_1= 53 000. a_n - Geometric sequence a_1= 53 000, r= 1.06 We want to find what will Kyla's salary be in four years. To do so, we will calculate a_4. In order to determine a_4 by using the formula for the nth term of a geometric sequence. a_n= a_1 r^(n-1) Let's substitute the values into the formula and find the value of a_4.

a_n=a_1 r^(n-1)

SubstituteValues

Substitute values

a_4= 53 000( 1.06)^(4-1)

▼

Simplify right-hand side

SubTerm

Subtract term

a_4=53 000(1.06)^3

CalcPow

Calculate power

a_4=53 000(1.191016)

Multiply

a_4=63 123.848

RoundInt

Round to nearest integer

a_4≈ 63 124

In four years, Kyla's salary will be about $63 124.

This time we want to find what will Kyla's salary be in 10 years. We can use the same geometric sequence that we found in Part A.

a_n - Geometric sequence a_1= 53 000, r= 1.06To determine a_(10), we will once again use the formula for the nth term of a geometric sequence. a_n= a_1 r^(n-1) Let's substitute the values into the formula and find the value of a_(10).