Advertisements
Advertisements
प्रश्न
The 4th term of a G.P. is square of its second term, and the first term is − 3. Find its 7th term.
Advertisements
उत्तर
\[\text { Let r be the common ratio of the given G . P } . \]
\[\text { Then }, a_4 = \left( a_2 \right)^2 \left[ \text { Given } \right]\]
\[\text { Now, }a r^3 = a^2 r^2 \]
\[ \Rightarrow r = a \]
\[ \Rightarrow r = - 3 \left[ \text { Putting } a = - 3 \right] \]
\[ \therefore a_7 = a r^6 \]
\[ \Rightarrow a_7 = \left( - 3 \right) \left( - 3 \right)^6 \left[ \text { Putting a = - 3 and }r = - 3 \right] \]
\[ \Rightarrow a_7 = \left( - 3 \right)\left( - 729 \right) \]
\[ \Rightarrow a_7 = - 2187\]
\[\text { Thus, the } 7^{th} \text { term of the G . P . is } - 2187 .\]
APPEARS IN
संबंधित प्रश्न
If the 4th, 10th and 16th terms of a G.P. are x, y and z, respectively. Prove that x, y, z are in G.P.
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`.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
4, −2, 1, −1/2, ...
Find :
the 10th term of the G.P.
\[\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, . . .\]
If 5th, 8th and 11th terms of a G.P. are p. q and s respectively, prove that q2 = ps.
Find the sum of the following geometric series:
`sqrt7, sqrt21, 3sqrt7,...` to n terms
Find the sum of the following serie:
5 + 55 + 555 + ... to n terms;
How many terms of the series 2 + 6 + 18 + ... must be taken to make the sum equal to 728?
The common ratio of a G.P. is 3 and the last term is 486. If the sum of these terms be 728, find the first term.
Show that the ratio of the sum of first n terms of a G.P. to the sum of terms from (n + 1)th to (2n)th term is \[\frac{1}{r^n}\].
A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying the odd places. Find the common ratio of the G.P.
Find the sum of the following serie to infinity:
\[1 - \frac{1}{3} + \frac{1}{3^2} - \frac{1}{3^3} + \frac{1}{3^4} + . . . \infty\]
Find the sum of the following serie to infinity:
8 + \[4\sqrt{2}\] + 4 + ... ∞
Find the sum of the following series to infinity:
10 − 9 + 8.1 − 7.29 + ... ∞
If a, b, c are in G.P., prove that log a, log b, log c are in A.P.
If a, b, c are in G.P., prove that the following is also in G.P.:
a2, b2, c2
If the 4th, 10th and 16th terms of a G.P. are x, y and z respectively. Prove that x, y, z are in G.P.
Given that x > 0, the sum \[\sum^\infty_{n = 1} \left( \frac{x}{x + 1} \right)^{n - 1}\] equals
In a G.P. of even number of terms, the sum of all terms is five times the sum of the odd terms. The common ratio of the G.P. is
In a G.P. if the (m + n)th term is p and (m − n)th term is q, then its mth term is
Mark the correct alternative in the following question:
Let S be the sum, P be the product and R be the sum of the reciprocals of 3 terms of a G.P. Then p2R3 : S3 is equal to
Check whether the following sequence is G.P. If so, write tn.
7, 14, 21, 28, …
For the G.P. if a = `7/243`, r = 3 find t6.
For the following G.P.s, find Sn
0.7, 0.07, 0.007, .....
For the following G.P.s, find Sn.
`sqrt(5)`, −5, `5sqrt(5)`, −25, ...
If S, P, R are the sum, product, and sum of the reciprocals of n terms of a G.P. respectively, then verify that `["S"/"R"]^"n"` = P2
Find: `sum_("r" = 1)^10 5 xx 3^"r"`
A ball is dropped from a height of 10m. It bounces to a height of 6m, then 3.6m and so on. Find the total distance travelled by the ball
Find GM of two positive numbers whose A.M. and H.M. are 75 and 48
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 a, b, c are in G.P. and ax2 + 2bx + c = 0 and px2 + 2qx + r = 0 have common roots then verify that pb2 – 2qba + ra2 = 0
Answer the following:
If p, q, r, s are in G.P., show that (p2 + q2 + r2) (q2 + r2 + s2) = (pq + qr + rs)2
In a G.P. of positive terms, if any term is equal to the sum of the next two terms. Then the common ratio of the G.P. is ______.
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 ______.
The lengths of three unequal edges of a rectangular solid block are in G.P. The volume of the block is 216 cm3 and the total surface area is 252cm2. The length of the longest edge is ______.
Find a G.P. for which sum of the first two terms is – 4 and the fifth term is 4 times the third term.
The sum of the infinite series `1 + 5/6 + 12/6^2 + 22/6^3 + 35/6^4 + 51/6^5 + 70/6^6 + ....` is equal to ______.
If `e^((cos^2x + cos^4x + cos^6x + ...∞)log_e2` satisfies the equation t2 – 9t + 8 = 0, then the value of `(2sinx)/(sinx + sqrt(3)cosx)(0 < x ,< π/2)` is ______.
The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in ______.
