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


Answer:

^@ 7^{ n } ^@

Step by Step Explanation:
  1. 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} ^@
  2. 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 ^@
  3. 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}@^
  4. Hence, the ^@ n^{th} ^@ term of the given ^@ G.P.^@ is ^@ 7^{ n } ^@.

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