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General term of Geometric ProgressionThe general term of geometric progression with first term 'a' and the common ratio is 'r' is given by$a_{n} = a r^{n 1}$ Where 'n' is a number of terms. Theorem : Prove that the nth term of G.P. with first term 'a' and the common ratio 'r' is given by $a_{n} = a r^{n 1}$ Proof: Let $a_{1},a_{2},a_{3},a_{4},...,a_{n}$ be the given G.P. then $a_{1}$ = a ⇒ a $r^{1 1}$ = a$r^{0}$ = a As 'r' is a common ratio ∴ $\frac{a_{2}}{a_{1}}$ = r ⇒ $a_{2} = a_{1} $r⇒ $a_{2}$ = ar $\frac{a_{3}}{a_{2}} = r ⇒ a_{3} = a_{2} r ⇒ a_{3} = (ar)r = ar^{2}=ar^{31}$ $\frac{a_{4}}{a_{3}} = r ⇒ a_{4} = a_{3} r⇒ a_{4} = (ar^{2})r = ar^{3} = ar^{41}$ Continuing in this manner we get , $a_{n} = ar^{n  1}$ So the sequence will be a, ar, a$r^{2}, ,....,ar^{n1}$ according as it is finite or infinite. Examples on general term of geometric progressionExample 1 : Write the general term of geometric progression if the sequence is 1,4,16,64,...Solution : General term is given by $a_{n} = ar^{n  1}$ (1) According to the given sequence first term = a = 1 ; common ratio = r = 4 ∴ equation (1) ⇒ $a_{n} = 1 \times 4^{n 1}$ $a_{n} = 4^{n 1}$ Example 2 : Write the general term of geometric progression if the sequence is $\frac{1}{4} , \frac{1}{2}$,1, 2, ... Solution : General term is given by $a_{n} = ar^{n  1}$ (1) According to the given sequence first term = a = $\frac{1}{4}$ ; common ratio = r = 2 ∴ equation (1) ⇒ $a_{n} = \frac{1}{4} \times (2)^{n 1}$ $a_{n} = 2^{2} \times (2)^{n 1}$ $a_{n} = 2^{2} \times 2^{n1} \times(1)^{n1}$ $a_{n} = (1)^{n 1} 2^{n 3}$ Example 3 : Write the general term of geometric progression if the sequence is 3,6,12,24, ... Solution : General term is given by $a_{n} = ar^{n  1}$ (1) According to the given sequence first term = a = 3 ; common ratio = r = 2 ∴ equation (1) ⇒ $a_{n} =3 \times 2^{n 1}$ $a_{n} = 3 \times 2^{n 1}$ From general term of geometric progression to Home Covid19 has led the world to go through a phenomenal transition . Elearning is the future today. Stay Home , Stay Safe and keep learning!!! Covid19 has affected physical interactions between people. Don't let it affect your learning.
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