Posted on - 08-02-2017

IIT JEE

A G.P. is a sequence whose first term is non-zero and each of whose succeeding term is r times the preceding term, where r is some fixed non - zero number, known as the common ratio of the G.P. .

If a
is the first term and r the common ratio, then G.P. can be written as a, ar,
ar^{2}, . . . the nth term, a_{n}, is given by a_{n}
= ar^{n-1}. The sum S_{n} of the first n terms of the G.P. is .

If -1 < r <
1, then the sum of the infinite G.P. a + ar + ar^{2} +........=

Notes:

- If each term of a G.P. is multiplied (divided) by a fixed non-zero constant, then the resulting sequence is also a G.P. with same common ratio as that of the given G.P.
- If each term of a
G.P. (with common ratio r) is raised to the power k, then the resulting
sequence is also a G.P. with common ratio r
^{k}. . - If a
_{1}, a_{2}, a_{3}, .......and b_{1}, b_{2}, b_{3}, ....... are two G.P.’s with common ratios r and r¢ respectively then the sequence a_{1}b_{1}, a_{2}b_{2}, a_{3}b_{3}.....is also a G.P. with common

ratio r r¢. - If we have to take three terms in a G.P., it is convenient to take them as a/r, a, ar. In general, we take in case we have to take (2k+1) terms in a G.P.
- If we have to take
four terms in a G.P., it is convenient to take them as a/r
^{3}, a/r, ar, ar^{3}. In general, we take , in case we have to take 2k terms in a G.P. - If a
_{1}, a_{2}, . . . . ,a_{n}are in G.P. , then a_{1}a_{n}= a_{2 }a_{n-1}= a_{3 }a_{n-2}= . . . . . - If a
_{1}, a_{2}, a_{3}, ….… is a G.P. ( each a_{I}> 0), then loga_{1}, loga_{2}, loga_{3 }….. is an A.P. The converse is also true.

- 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 = is the geometric mean of a and c.
- If a
_{1}, a_{2}......a_{n}are non-zero positive numbers then their G.M (G) is given by

G = (a_{1}a_{2}a_{3}......a_{n})^{1/n}. If G_{1}, G_{2},…..G_{n}are n geometric means between a and b then

a, G_{1}, G_{2}, …., G_{n}, b will be a G.P. Here b = a r^{n + 1}.

&⇒ r =

&⇒, G_{r}= awhere G_{r}is the r^{th }mean.

The third term of a G.P. is 7. Find the product of first five terms.

Let the terms be , a , ar, ar^{2
}

&⇒ a = 7.

The product = a^{5}
= 7^{5}.

Show that is a composite number.

We have = 10^{90} +10^{89} + 10^{88}
+ . . . … . + 10^{2} + 10 +1

= = = m ´ n , where m and n are
natural numbers ( a^{n}-1 is divisible by a-1) ,

Hence is a composite number.