### Find the ^@ n^{th} ^@ term of the ^@G.P. 7, 49, 343, .......^@

**Answer:**

^@ 7^{ n } ^@

**Step by Step Explanation:**

- A geometric progression ^@(G.P.)^@ is of the form, ^@a, ar, ar^2, ar^3, ......, ^@ where ^@a^@ is called the first term and ^@r^@ is called the common ratio of the ^@G.P.^@

The ^@n^{ th }^@ term of a ^@G.P.^@ is given by, ^@a_n= ar^{n-1} ^@ - Here, the first term, ^@a = 7^@

The common ratio, ^@r = \dfrac{ a_{k+1} }{ a_k } ^@ where ^@ k \ge 1 ^@

^@ \implies r = \dfrac{a_{1+1} }{ a_1 } = \dfrac{ a_2 }{ a_1 } = \dfrac{ 49 }{ 7 } = 7 ^@ - Now, we need to find the ^@ n^{th} ^@ term of the ^@G.P., i.e. a_n.^@

@^ \begin{align} & a_{ n } = ar^{ n-1 } \\ \implies & a_n = 7(7)^{ n-1 } \\ \implies & a_n = 7 ^{ 1 + n-1 } \\ \implies & a_n = 7^n \end{align}@^ - Hence, the ^@ n^{th} ^@ term of the given ^@ G.P.^@ is ^@ 7^{ n } ^@.