Advertisements
Advertisements
प्रश्न
The sum or difference of two G.P.s, is again a G.P.
विकल्प
True
False
Advertisements
उत्तर
This statement is False.
Explanation:
Let us consider two G.P.’s
a1, a1r1, a1r12, a1r13 ... a1r1n-1
And a2, a2r2, a2r22, a2r23, ... a2r2n-1
Now Sum of two G.Ps
`(a_1 + a_2) + (a_1r_1 + a_2r_2) + (a_1r_1^2 + a_2r_2^2) ...`
Now `T_2/T_1 = (a_1r_1 + a_2r_2)/(a_1 + a_2)`
And `T_3/T_2 = (a_1r_1^2 + a_2r_2^2)/(a_1r_1 + a_2r_2)`
But `(a_1r_1 + a_2r_2)/(a_1 + a_2) ≠ (a_1r_1^2 + a_2r_2^2)/(a_1r_1 + a_2r_2)`
Now let us consider the difference G.P’s
`(a_1 - a_2) + (a_1r_1 - a_2r_2) + (a_1r_1^2 - a_2r_2^2)`
∴ `T_2/T_1 = (a_1r_1 - a_2r_2)/(a_1 - a_2)`
And `T_3/T_2 = (a_1r_1^2 - a_2r_2^2)/(a_1r_1 - a_2r_2)`
But `T_2/T_1 ≠ T_3/T_2`
APPEARS IN
संबंधित प्रश्न
Find the 20th and nthterms of the G.P. `5/2, 5/4 , 5/8,...`
The 4th term of a G.P. is square of its second term, and the first term is –3. Determine its 7thterm.
How many terms of G.P. 3, 32, 33, … are needed to give the sum 120?
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`
Find the value of n so that `(a^(n+1) + b^(n+1))/(a^n + b^n)` may be the geometric mean between a and b.
The sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio `(3 + 2sqrt2) ":" (3 - 2sqrt2)`.
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:
−2/3, −6, −54, ...
Which term of the G.P. :
\[2, 2\sqrt{2}, 4, . . .\text { is }128 ?\]
If the pth and qth terms of a G.P. are q and p, respectively, then show that (p + q)th term is \[\left( \frac{q^p}{p^q} \right)^\frac{1}{p - q}\].
The sum of three numbers in G.P. is 14. If the first two terms are each increased by 1 and the third term decreased by 1, the resulting numbers are in A.P. Find the numbers.
How many terms of the G.P. 3, 3/2, 3/4, ... be taken together to make \[\frac{3069}{512}\] ?
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 \[\frac{1}{r^n}\].
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.
Show that in an infinite G.P. with common ratio r (|r| < 1), each term bears a constant ratio to the sum of all terms that follow it.
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:
a (b2 + c2) = c (a2 + b2)
If a, b, c are in G.P., prove that:
\[a^2 b^2 c^2 \left( \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \right) = a^3 + b^3 + c^3\]
Insert 6 geometric means between 27 and \[\frac{1}{81}\] .
Insert 5 geometric means between 16 and \[\frac{1}{4}\] .
If a = 1 + b + b2 + b3 + ... to ∞, then write b in terms of a.
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
Check whether the following sequence is G.P. If so, write tn.
3, 4, 5, 6, …
Find three numbers in G.P. such that their sum is 21 and sum of their squares is 189.
The numbers 3, x, and x + 6 form are in G.P. Find 20th term.
For the following G.P.s, find Sn.
`sqrt(5)`, −5, `5sqrt(5)`, −25, ...
For a G.P. a = 2, r = `-2/3`, find S6
If one invests Rs. 10,000 in a bank at a rate of interest 8% per annum, how long does it take to double the money by compound interest? [(1.08)5 = 1.47]
Express the following recurring decimal as a rational number:
`2.bar(4)`
Answer the following:
Find `sum_("r" = 1)^"n" (2/3)^"r"`
Answer the following:
Find k so that k – 1, k, k + 2 are consecutive terms of a G.P.
Answer the following:
If p, q, r, s are in G.P., show that (p2 + q2 + r2) (q2 + r2 + s2) = (pq + qr + rs)2
Answer the following:
Find the sum of infinite terms of `1 + 4/5 + 7/25 + 10/125 + 13/6225 + ...`
If a, b, c, d are in G.P., prove that a2 – b2, b2 – c2, c2 – d2 are also in G.P.
If a, b, c, d are four distinct positive quantities in G.P., then show that a + d > b + c
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 P2 R3 : S3 is equal to ______.
If the pth and qth terms of a G.P. are q and p respectively, show that its (p + q)th term is `(q^p/p^q)^(1/(p - q))`
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 ______.
