Question

If the first term of a G.P be 5 and the common ratio is -5, then which term is 3125
A. 6th
B. 5th
C. 7th
D.8th

Answer 1

Hint: G.P or geometric progression is a sequence that increases or decreases, where any of the terms divided by the adjacent previous term of the series gives a ratio that is a common ratio for all the terms. The ratio and the terms can be positive or negative. If we know the first term and the common ratio of G.P. we can multiply the adjacent terms with the common ratio till the term that we want.


Formula Used: \[{T_n} = a{r^{n - 1}}\], where \[{T_n}\] is the nth term of the geometric progression, a is the first term of the series and r is the common ratio of the series.


Complete step by step answer: We have been given that a = 5 is the first term of a G.P and r = -5 is the common ratio. We have been also given the nth term as \[{T_n} = 3125\].
So, using the formula of the nth term of geometric progression, we can find the value of n as,
\[
  {T_n} = a{r^{n - 1}} \\
   \Rightarrow 3125 = 5 \times {\left( { - 5} \right)^{n - 1}} \\
   \Rightarrow - 3125 = {\left( { - 5} \right)^1} \times {\left( { - 5} \right)^{n - 1}} \\
   \Rightarrow {\left( { - 5} \right)^5} = {\left( { - 5} \right)^{\left( {n - 1} \right) + 1}}
 \]

Further Simplifying, we get,
\[ \Rightarrow {\left( { - 5} \right)^5} = {\left( { - 5} \right)^n}\]
As the base of both sides is equal, we can directly equate the powers, which gives us the result as, n = 5.

So, option B, 5th term is the required solution

Note: When we are finding the nth term of a geometric progression, remember that to find the nth term of the progression, the power of the common ratio in the formula is (n – 1). While finding what power of -5 is -3125, remember that the power cannot be even otherwise the result will be positive. Only odd powers of a negative base give the result as a negative number.