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- If â€˜aâ€™ is the first term and â€˜râ€™ is the common ratio of the geometric progression, then its n
^{th}term is given by an = ar^{n-1 } - The sum, S
_{n}of the first â€˜nâ€™ terms of the G.P. is given by S_{n}= a (r^{n}â€“ 1)/ (r-1), when r â‰ 1; = na , if r =1 - If -1 < x < 1, then limx
^{n}= 0, as n â†’âˆž. Hence, the sum of an infinite G.P. is 1+x+x^{2}+ â€¦.. = 1/(1-x) - If -1 < r< 1, then the sum of the infinite G.P. is a +ar+ ar
^{2}+ â€¦.. = a/(1-r) - If each term of the G.P is multiplied or divided by a non-zero fixed constant, the resulting sequence is again a G.P.
- If a
_{1}, a_{2}, a_{3}, â€¦. and b_{1}, b_{2}, b_{3}, â€¦ are two geometric progressions, then a_{1}b_{1}, a2b_{2}, a_{3}b_{3}, â€¦â€¦ is also a geometric progression and a_{1}/b_{1}, a_{2}/b_{2}, ... ... ..., a_{n}/b_{n }will also be in G.P. - Suppose a
_{1}, a_{2}, a_{3}, â€¦â€¦,an are in G.P. then a_{n}, a_{nâ€“1}, a_{nâ€“2}, â€¦â€¦, a_{3}, a_{2}, a_{1}will also be in G.P. - Taking the inverse of a G.P. also results a G.P. Suppose a
_{1}, a_{2}, a_{3}, â€¦â€¦,an are in G.P then 1/a_{1}, 1/a_{2}, 1/a_{3}â€¦â€¦, 1/an will also be in G.P - If we need to assume three numbers in G.P. then they should be assumed as a/b, a, ab (here common ratio is b)
- Four numbers in G.P. should be assumed as a/b
^{3}, a/b, ab, ab^{3}(here common ratio is b^{2}) - Five numbers in G.P. a/b
^{2}, a/b, a, ab, ab^{2}(here common ratio is b) - If a
_{1}, a_{2}, a_{3},â€¦ ,an is a G.P (a_{i}> 0 âˆ€i), then log a_{1}, log a_{2}, log a_{3}, â€¦â€¦, log an is an A.P. In this case, the converse of the statement also holds good. - If three terms are in G.P., then the middle term is called the geometric mean (G.M.) between the two. So if a, b, c are in G.P., then b = âˆšac is the geometric mean of a and c.
- Likewise, if a
_{1}, a_{2}, â€¦â€¦,an are non-zero positive numbers, then their G.M.(G) is given by G = (a_{1}a_{2}a_{3}â€¦â€¦ a_{n})^{1/n}. - If G
_{1}, G_{2}, â€¦â€¦G_{n }are n geometric means between and a and b then a, G_{1}, G_{2}, â€¦â€¦,G_{n}b will be a G.P. - Here b = ar
^{n+1}, â‡’ r =^{n+1}âˆšb/a, Hence, G_{1}= a.^{n+1}âˆšb/a, G_{2 }= a(^{n+1}âˆšb/a)^{2},â€¦, G_{n}= a(^{n+1}âˆšb/a)^{n}.

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