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Series: Geometric Progression Formula Manipulation
If the sum of n terms of a geometric series with
common ratio
1
---
2
 starting at
1
---
5
 is 0.399609375 then
n must be

tickcross


Worked Solution:

The sum, S of a geometric series of n terms
starting at a with common ratio r is given by S =
a (1 – rn)
---
1 – r
The questions asks for n:
rn = 1 –
S (1 – r)
---
a
Taking logs of both sides:
n log r = log(1 –
S (1 – r)
---
a
)
Putting the values in the question into this formula we have:
n log
1
---
2
 = log(1 –
0.399609 (1 –
1
–--
2
)
---
1
–--
5
)
=>n log 0.5 = log 0.000977
Using logs to base 10, the number of terms to make a sum of 0.399609 is
n =
log10 0.000977
---
log10 0.5
 =
–3.0103
---
–0.30103
 = 10
=>n = 10

© MEI Produced by Dr Ron Knott, revised 12 November 2007
tom . button [AT] mei . org . uk 		Test reference: SeriesGPFormula.html?ref=34166,qu=finn