Advertisements
Advertisements
प्रश्न
Show that one of the following progression is a G.P. Also, find the common ratio in case:
−2/3, −6, −54, ...
Advertisements
उत्तर
We have,
\[ a_1 = \frac{- 2}{3} , a_2 = - 6, a_3 = - 54\]
\[\text { Now }, \frac{a_2}{a_1} = \frac{- 6}{\frac{- 2}{3}} = 9, \frac{a_3}{a_2} = \frac{- 54}{- 6} = 9 \]
\[ \therefore \frac{a_2}{a_1} = \frac{a_3}{a_2} = 9\]
\[\text { Thus, } a_1 , a_2 \text { and } a_3 \text { are in G . P . , where } a = \frac{- 2}{3}\text { and } r = 9 .\]
संबंधित प्रश्न
Which term of the following sequence:
`1/3, 1/9, 1/27`, ...., is `1/19683`?
Evaluate `sum_(k=1)^11 (2+3^k )`
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 a, b, c, d are in G.P, prove that (an + bn), (bn + cn), (cn + dn) are in G.P.
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 4th term from the end of the G.P.
\[\frac{1}{2}, \frac{1}{6}, \frac{1}{18}, \frac{1}{54}, . . . , \frac{1}{4374}\]
In a GP the 3rd term is 24 and the 6th term is 192. Find the 10th term.
If a, b, c, d and p are different real numbers such that:
(a2 + b2 + c2) p2 − 2 (ab + bc + cd) p + (b2 + c2 + d2) ≤ 0, then show that a, b, c and d are in G.P.
Find three numbers in G.P. whose product is 729 and the sum of their products in pairs is 819.
Find the sum of the following geometric progression:
4, 2, 1, 1/2 ... to 10 terms.
Find the sum of the following geometric series:
x3, x5, x7, ... to n terms
Find the sum of the following series:
9 + 99 + 999 + ... to n terms;
Find the sum of the following series:
0.6 + 0.66 + 0.666 + .... to n terms
How many terms of the series 2 + 6 + 18 + ... must be taken to make the sum equal to 728?
The 4th and 7th terms of a G.P. are \[\frac{1}{27} \text { and } \frac{1}{729}\] respectively. Find the sum of n terms of the G.P.
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 series to infinity:
`1/3+1/5^2 +1/3^3+1/5^4 + 1/3^5 + 1/56+ ...infty`
If Sp denotes the sum of the series 1 + rp + r2p + ... to ∞ and sp the sum of the series 1 − rp + r2p − ... to ∞, prove that Sp + sp = 2 . S2p.
Find the sum of the terms of an infinite decreasing G.P. in which all the terms are positive, the first term is 4, and the difference between the third and fifth term is equal to 32/81.
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, 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)
Find the geometric means of the following pairs of number:
−8 and −2
The fractional value of 2.357 is
If pth, qth and rth terms of an A.P. are in G.P., then the common ratio of this G.P. is
Check whether the following sequence is G.P. If so, write tn.
`sqrt(5), 1/sqrt(5), 1/(5sqrt(5)), 1/(25sqrt(5))`, ...
Check whether the following sequence is G.P. If so, write tn.
3, 4, 5, 6, …
For the G.P. if r = `1/3`, a = 9 find t7
The number of bacteria in a culture doubles every hour. If there were 50 bacteria originally in the culture, how many bacteria will be there at the end of 5th hour?
The numbers x − 6, 2x and x2 are in G.P. Find 1st term
Express the following recurring decimal as a rational number:
`0.bar(7)`
The sum of 3 terms of a G.P. is `21/4` and their product is 1 then the common ratio is ______.
Answer the following:
For a sequence , if tn = `(5^("n" - 2))/(7^("n" - 3))`, verify whether the sequence is a G.P. If it is a G.P., find its first term and the common ratio.
Answer the following:
Find `sum_("r" = 1)^"n" (2/3)^"r"`
Answer the following:
Which 2 terms are inserted between 5 and 40 so that the resulting sequence is G.P.
For a, b, c to be in G.P. the value of `(a - b)/(b - c)` is equal to ______.
If 0 < x, y, a, b < 1, then the sum of the infinite terms of the series `sqrt(x)(sqrt(a) + sqrt(x)) + sqrt(x)(sqrt(ab) + sqrt(xy)) + sqrt(x)(bsqrt(a) + ysqrt(x)) + ...` is ______.
If the expansion in powers of x of the function `1/((1 - ax)(1 - bx))` is a0 + a1x + a2x2 + a3x3 ....... then an is ______.
