In his first year, a math teacher earned $32,000. Each successive year, he

earned a 5% raise. How much did he earn in his 20th year? What were his total

earnings over the 20-year period?

Explain steps by step

Step-by-step explanation:

Each successive year, he

earned a 5% raise. It means that the salary is increasing in geometric progression. The formula for determining the nth term of a geometric progression is expressed as

Tn = ar^ (n - 1)

Where

a represents the first term of the sequence (amount earned in the first year).

r represents the common ratio.

n represents the number of terms (years).

From the information given,

a = $32,000

r = 1 + 5/100 = 1.05

n = 20 years

The amount earned in his 20th year, T20 is

T20 = 32000 * 1.05^ (20 - 1)

T20 = 32000 * 1.05^ (19)

T20 = $80862.4

To determine the his total

earnings over the 20-year period, we would apply the formula for determining the sum of n terms, Sn of a geometric sequence which is expressed as

Sn = (ar^n - 1) / (r - 1)

Therefore, the sum of the first 20 terms, S20 is

S20 = (32000 * 1.05^ (20) - 1) / 1.05 - 1

S20 = (32000 * 1.653) / 0.05

S20 = $1057920