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Which term of the G.P. 5, 10, 20, 40, ..... is 5120 ?

Answer
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Hint: Apply the basic formula for finding the nth terms of G.P. i.e. Tn=arn1 where ‘a’ is the first term and ‘r’ is the common ratio of the G.P.

Complete step-by-step answer:
The given GP series is
5, 10, 20, 40, ............................(1)
We can define any GP series as
A,AR,AR2,AR3...........ARn1
Where A = first term
R = common ratio
Here, next coming term is ‘R’ times of previous terms;
Hence, the common ratio is the ratio between any two continuous terms.
Now, series from equation (1) is;
5, 10, 20, 40..........
Hence, first term is 5 and common Ratio=105=2010=4020=2 from the basic definitions of geometric progression.
Therefore a = first term of given series = 5
And r = common ratio = 2.
Now, we have to determine which term will be 5120 in the given series.
As, we have general term or nth terms of G.P. is given in terms of first term and common ratio (r) as
an=arn1...............(2)
Now, we can observe from the question and equation (1) that term 5120 is a term of G.P. and we have to determine ‘n’ i.e. at what number it will come in series.
Hence, using equation (1), we have
an=5120r=2a=5n=?
Putting all the values to equation (1), we get
5120=5(2)n1
Transferring 5 to other side, we get
(2)n1=51205=1024(2)n1=1024
Now, we can factorize 1024 and it can be written as 210. Hence, above equation can be simplified as
(2)n1=(2)10
As, bases of the surds are equal, hence power should also be equal to get the same result of the whole surd.
Hence,
(n1)=10n=11
So, we get that 5120 be the 11th term of the given G.P.
Hence, the answer is 11.
Note: One can go wrong while applying nth term formula of G.P. One can put n = 5120 and try to calculate an=arn1 which is wrong. So understanding the parameters used in an=arn1is also an important side of the question.
One can go wrong by applying formula of nthterm of A.P i.e. an=a+(n1)d which is wrong. As the given series is G.P. So, we cannot use any property of series A.P.
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