Sum of first five terms is \(19.3785\) OR in IMPROPER \(\frac{155}{8}\).
What is a geometric sequence?
- When the ratio of consecutive terms of a sequence has a common ratio then it is a geometric sequence.
- In other words, each term is found by multiplying the previous term with a constant number called the common ratio.
How to find the common ratio of a geometric sequence
- As each term is found by multiplying the preceding term by the common ratio r, the common ratio can be found using the ratio of two consecutive terms which is
- \(r= \frac{a_{n} }{a_{n} -1}\)
Geometric series formula
- When the terms of a geometric sequence are added then it is called geometric series.
- If \(a_{n} = a_{1} . r^{n-1} , r
eq 1\) than the sum of the first n terms of the sequence can be found using the formula
\(S_{n} = a_{1} . (\frac{1-r^{n} }{1-r } )\)
So according the question
Given :
First term \(a1 = 10\)
common ratio \(r= 1/2\) \(= 0.5\)
number of term \(n = 5\)
Sum of the first five terms \(S_{n} = a_{1} . (\frac{1-r^{n} }{1-r } )\) ,
\(S_{5} =\frac{10.(1 - 0.5^{5} )}{1 - 0.5}\) = \(19.375\)
Hence , the sum of the first five terms of the geometric sequence is \(19.3785\) or in improper \(\frac{155}{8}\).
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