SOLUTION: sum of a geometric series is 46.5. The first term of the series is 24​, and its common ratio is 0.5. How many terms are there in the​ series?

Algebra ->  Sequences-and-series -> SOLUTION: sum of a geometric series is 46.5. The first term of the series is 24​, and its common ratio is 0.5. How many terms are there in the​ series?      Log On


   

Question 1156304: sum of a geometric series is 46.5. The first term of the series is 24​, and its common ratio is 0.5. How many terms are there in the​ series?
Found 2 solutions by jim_thompson5910, MathTherapy:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

n = number of terms = unknown
S = 46.5 = sum of the first n terms
a = 24 = first term
r = 0.5 = common ratio

S+=+%28a%281-r%5En%29%29%2F%281-r%29 Formula used to add the first n terms of a geometric sequence

46.5+=+%2824%281-0.5%5En%29%29%2F%281-0.5%29 Substitution

46.5+=+%2824%281-0.5%5En%29%29%2F0.5

46.5%2A0.5+=+24%281-0.5%5En%29 Multiply both sides by 0.5

23.25+=+24%281-0.5%5En%29

24%281-0.5%5En%29+=+23.25

1-0.5%5En+=+23.25%2F24 Divide both sides by 24

1-0.5%5En+=+0.96875

0.5%5En+=+1-0.96875 Isolate the term 0.5^n

0.5%5En+=+0.03125

log%28%280.5%5En%29%29+=+log%28%280.03125%29%29 Apply logs to both sides

n%2Alog%28%280.5%29%29+=+log%28%280.03125%29%29 We do so to be able to pull the exponent n down.

n+=+log%28%280.03125%29%29%2Flog%28%280.5%29%29 Divide both sides by log(0.5)

n+=+5 Use a calculator

Answer: 5 terms

The first five terms are: {24, 12, 6, 3, 1.5}
Each new term is found by multiplying the previous term by the common ratio 0.5

Those five terms add to
24 + 12 + 6 + 3 + 1.5 = 46.5
which confirms our answer.

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!

sum of a geometric series is 46.5. The first term of the series is 24​, and its common ratio is 0.5. How many terms are there in the​ series?
Formula for sum of a G.P.: 

                           matrix%281%2C3%2C+46.5%2C+%22=%22%2C+48%281+-+.5%5En%29%29

                           matrix%281%2C3%2C+46.5%2F48%2C+%22=%22%2C+1+-+.5%5En%29

                           matrix%281%2C3%2C+.5%5En%2C+%22=%22%2C+1+-+46.5%2F48%29

                           Number of terms, or