# A woman made 35,000 during her first year as a dog trainer. Each year she has recieved a 10% raise. Find the total earnings if this is her 8th year. (Use geometric sequence/Partial sum formula if...

A woman made 35,000 during her first year as a dog trainer. Each year she has recieved a 10% raise. Find the total earnings if this is her 8th year. (Use geometric sequence/Partial sum formula if you can)

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The initial value a of the sequence is 35,000, the ratio is 1.1 since every year she receives the amount of the previous year plus 10%. The number of terms m is 8.

`a=35,000`

`r=1.1`

`m=8`

The sum of the first m members is equal to `(a(1-r^m))/(1-r)`

Substitute 35,000 for a 1.1 for r and 8 for m to determine the total earnings for the first 8 years.

`(35,000(1-1.1^8))/(1-1.1)`

`=(35,000(-1.1436))/(-0.1)`

`=400,256`

**Thus she earned $400,255 after 8 years.**

` ` To verify, write each term of the sequence and add them together:

n=1, 35,000

n=2, 35,000*1.1=38,500

n=3, 38,500*1.1=42,350

n=4, 42,350*1.1=46,585

n=5, 46,585*1.1=51,243.50

n=6, 51,243.50*1.1=56,367

n=7, 56,367*1.1=62,004.6

n=8, 62,004.6*1.1=68,205

**The sum of all of them is 400,255**

A woman made 35,000 during her first year as a dog trainer. Her salary increases by 10% every year.

As a geometric series, the first term of the series is a = 35000 and the common ratio is r = 1.1

The sum of eight years of her earnings is the sum of the first 8 terms of the geometric series with first term 35000 and common ratio 1.1.

This is equal to 35000*(1.1^8 - 1)/(1.1 - 1)

= 35000*(1.1^8 - 1)/(1.1 - 1)

~~ 400256

The woman's total salary is approximately $400256