Advertisements
Advertisements
प्रश्न
If the fifth term of a G.P. is 2, then write the product of its 9 terms.
Advertisements
उत्तर
Here, a5 = 2
\[\Rightarrow a r^4 = 2\]
Product of the nine terms, i.e.
\[a, ar, a r^2 , a r^3 , a r^4 , a r^5 , a r^6 , a r^7 \text { and } a r^8\]
\[\left( a \times a r^8 \right)\left( ar \times a r^7 \right)\left( a r^2 \times a r^6 \right)\left( a r^3 \times a r^5 \right)\left( a r^4 \right) = \left( a r^4 \right)^9 \]
\[ \because a r^4 = 2\]
\[\text { Required product } = 2^9 = 512\]
संबंधित प्रश्न
If the pth, qth and rth terms of a G.P. are a, b and c, respectively. Prove that `a^(q - r) b^(r-p) c^(p-q) = 1`.
If the first and the nth term of a G.P. are a ad b, respectively, and if P is the product of n terms, prove that P2 = (ab)n.
Insert two numbers between 3 and 81 so that the resulting sequence is G.P.
Which term of the G.P. :
\[2, 2\sqrt{2}, 4, . . .\text { is }128 ?\]
Find three numbers in G.P. whose sum is 65 and whose product is 3375.
Find three numbers in G.P. whose sum is 38 and their product is 1728.
The product of three numbers in G.P. is 216. If 2, 8, 6 be added to them, the results are in A.P. Find the numbers.
Find the sum of the following geometric progression:
2, 6, 18, ... to 7 terms;
Find the sum of the following geometric series:
\[\sqrt{2} + \frac{1}{\sqrt{2}} + \frac{1}{2\sqrt{2}} + . . .\text { to 8 terms };\]
Find the sum of the following geometric series:
`sqrt7, sqrt21, 3sqrt7,...` to n terms
How many terms of the series 2 + 6 + 18 + ... must be taken to make the sum equal to 728?
Prove that: (91/3 . 91/9 . 91/27 ... ∞) = 3.
Find the rational numbers having the following decimal expansion:
\[0 . 6\overline8\]
Find an infinite G.P. whose first term is 1 and each term is the sum of all the terms which follow it.
If a, b, c are in G.P., prove that:
\[\frac{1}{a^2 - b^2} + \frac{1}{b^2} = \frac{1}{b^2 - c^2}\]
If a, b, c, d are in G.P., prove that:
(a + b + c + d)2 = (a + b)2 + 2 (b + c)2 + (c + d)2
If a, b, c, d are in G.P., prove that:
(a2 + b2 + c2), (ab + bc + cd), (b2 + c2 + d2) are in G.P.
If (a − b), (b − c), (c − a) are in G.P., then prove that (a + b + c)2 = 3 (ab + bc + ca)
If (p + q)th and (p − q)th terms of a G.P. are m and n respectively, then write is pth term.
If logxa, ax/2 and logb x are in G.P., then write the value of x.
Write the product of n geometric means between two numbers a and b.
If a = 1 + b + b2 + b3 + ... to ∞, then write b in terms of a.
The value of 91/3 . 91/9 . 91/27 ... upto inf, is
If a, b, c are in G.P. and x, y are AM's between a, b and b,c respectively, then
Check whether the following sequence is G.P. If so, write tn.
`sqrt(5), 1/sqrt(5), 1/(5sqrt(5)), 1/(25sqrt(5))`, ...
The numbers 3, x, and x + 6 form are in G.P. Find x
For the following G.P.s, find Sn.
`sqrt(5)`, −5, `5sqrt(5)`, −25, ...
Find: `sum_("r" = 1)^10 5 xx 3^"r"`
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
9, 8.1, 7.29, ...
The sum of an infinite G.P. is 5 and the sum of the squares of these terms is 15 find the G.P.
Find : `sum_("n" = 1)^oo 0.4^"n"`
Insert two numbers between 1 and −27 so that the resulting sequence is a G.P.
Select the correct answer from the given alternative.
Sum to infinity of a G.P. 5, `-5/2, 5/4, -5/8, 5/16,...` is –
Answer the following:
Find the sum of the first 5 terms of the G.P. whose first term is 1 and common ratio is `2/3`
Answer the following:
If p, q, r, s are in G.P., show that (p2 + q2 + r2) (q2 + r2 + s2) = (pq + qr + rs)2
If x, 2y, 3z are in A.P., where the distinct numbers x, y, z are in G.P. then the common ratio of the G.P. is ______.
