Advertisements
Advertisements
प्रश्न
Answer the following:
Which 2 terms are inserted between 5 and 40 so that the resulting sequence is G.P.
Advertisements
उत्तर
Let the required numbers be G1 and G2.
∴ 5, G1, G2, 40 are in G.P.
∴ t1 = 5, t2 = G1, t3 = G2, t4 = 40
∴ t1 = a = 5, t4 = 40
tn = arn–1
∴ t4 = 5(r)4–1
∴ 40 = 5r3
∴ r3 = 8 = 23
∴ r = 2
G1 = t2 = ar = 5 (2) = 10
G2 = t3 = ar2 = 5(2)2 = 20
∴ For resulting sequence to be in G.P. we need to insert numbers 10 and 20.
APPEARS IN
संबंधित प्रश्न
The 5th, 8th and 11th terms of a G.P. are p, q and s, respectively. Show that q2 = ps.
Find the sum to indicated number of terms of the geometric progressions `sqrt7, sqrt21,3sqrt7`...n terms.
Find the sum to n terms of the sequence, 8, 88, 888, 8888… .
Find the sum of the products of the corresponding terms of the sequences `2, 4, 8, 16, 32 and 128, 32, 8, 2, 1/2`
If a, b, c are in A.P,; b, c, d are in G.P and ` 1/c, 1/d,1/e` are in A.P. prove that a, c, e are in G.P.
Which term of the progression 0.004, 0.02, 0.1, ... is 12.5?
The product of three numbers in G.P. is 125 and the sum of their products taken in pairs is \[87\frac{1}{2}\] . Find them.
The sum of three numbers in G.P. is 21 and the sum of their squares is 189. Find the numbers.
Find the sum of the following geometric series:
(x +y) + (x2 + xy + y2) + (x3 + x2y + xy2 + y3) + ... to n terms;
Find the sum of the following geometric series:
1, −a, a2, −a3, ....to n terms (a ≠ 1)
Find the sum of the following series:
0.5 + 0.55 + 0.555 + ... to n terms.
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 person has 2 parents, 4 grandparents, 8 great grandparents, and so on. Find the number of his ancestors during the ten generations preceding his own.
Find the sum of the following series to infinity:
10 − 9 + 8.1 − 7.29 + ... ∞
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.
If S denotes the sum of an infinite G.P. S1 denotes the sum of the squares of its terms, then prove that the first term and common ratio are respectively
\[\frac{2S S_1}{S^2 + S_1}\text { and } \frac{S^2 - S_1}{S^2 + S_1}\]
Find k such that k + 9, k − 6 and 4 form three consecutive terms of a G.P.
If a, b, c are in G.P., prove that the following is also in G.P.:
a2 + b2, ab + bc, b2 + c2
Find the geometric means of the following pairs of number:
−8 and −2
If pth, qth and rth terms of a G.P. re x, y, z respectively, then write the value of xq − r yr − pzp − q.
If S be the sum, P the product and R be the sum of the reciprocals of n terms of a GP, then P2 is equal to
If pth, qth and rth terms of an A.P. are in G.P., then the common ratio of this G.P. is
If second term of a G.P. is 2 and the sum of its infinite terms is 8, then its first term is
If x is positive, the sum to infinity of the series \[\frac{1}{1 + x} - \frac{1 - x}{(1 + x )^2} + \frac{(1 - x )^2}{(1 + x )^3} - \frac{(1 - x )^3}{(1 + x )^4} + . . . . . . is\]
If x = (43) (46) (46) (49) .... (43x) = (0.0625)−54, the value of x is
Check whether the following sequence is G.P. If so, write tn.
1, –5, 25, –125 …
Check whether the following sequence is G.P. If so, write tn.
7, 14, 21, 28, …
Find four numbers in G.P. such that sum of the middle two numbers is `10/3` and their product is 1
If p, q, r, s are in G.P. show that p + q, q + r, r + s are also in G.P.
The numbers 3, x, and x + 6 form are in G.P. Find x
Mosquitoes are growing at a rate of 10% a year. If there were 200 mosquitoes in the beginning. Write down the number of mosquitoes after 10 years.
For the following G.P.s, find Sn.
p, q, `"q"^2/"p", "q"^3/"p"^2,` ...
For a sequence, if Sn = 2(3n –1), find the nth term, hence show that the sequence is a G.P.
Find: `sum_("r" = 1)^10(3 xx 2^"r")`
If the first term of the G.P. is 16 and its sum to infinity is `96/17` find the common ratio.
Insert two numbers between 1 and −27 so that the resulting sequence is a G.P.
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
