Advertisements
Advertisements
प्रश्न
If Sn, S2n, S3n are the sum of n, 2n, 3n terms of a G.P. respectively, then verify that Sn (S3n – S2n) = (S2n – Sn)2.
Advertisements
उत्तर
Let a and r be the 1st term and common ratio of the G.P. respectively.
∴ Sn = `"a"(("r"^"n" - 1)/("r" - 1))`, S2n = `"a"(("r"^(2"n") - 1)/("r" - 1))`, S3n = `"a"(("r"^(3"n") - 1)/("r" - 1))`
∴ S2n – Sn = `"a"(("r"^(2"n") - 1)/("r" - 1)) - "a"(("r"^"n"- 1)/("r" - 1))`
= `"a"/("r" - 1)("r"^(2"n") - 1 - "r"^"n" + 1)`
= `"a"/("r" - 1)("r"^(2"n") - "r"^"n")`
= `"ar"^"n"/("r" - 1)("r"^"n" - 1)`
∴ S2n – Sn = `"r"^"n"*("a"("r"^"n" - 1))/("r" - 1)` ....(i)
S3n – S2n = `"a"(("r"^(3"n") - 1)/("r" - 1)) - "a"(("r"^(2"n") - 1)/("r" - 1))`
= `"a"/("r" - 1)("r"^(3"n") - 1 - "r"^(2"n") + 1)`
= `"a"/("r" - 1)("r"^(3"n") - "r"^(2"n"))`
= `"a"/("r" - 1)*"r"^(2"n")("r"^"n" - 1)`
= `"a"*(("r"^"n" - 1)/("r" - 1))*"r"^(2"n")`
∴ Sn(S3n – S2n) = `["a"*(("r"^"n"- 1)/("r" - 1))]["a"*(("r"^"n" - 1)/("r" - 1))"r"^(2"n")]`
= `["r"^"n"*("a"("r"^"n" - 1))/("r" - 1)]^2`
∴ Sn(S3n – S2n) = (S2n – Sn)2 ...[From (i)]
APPEARS IN
संबंधित प्रश्न
Find the 20th and nthterms of the G.P. `5/2, 5/4 , 5/8,...`
Find a G.P. for which sum of the first two terms is –4 and the fifth term is 4 times the third term.
Find:
the 10th term of the G.P.
\[- \frac{3}{4}, \frac{1}{2}, - \frac{1}{3}, \frac{2}{9}, . . .\]
Find the 4th term from the end of the G.P.
Which term of the progression 0.004, 0.02, 0.1, ... is 12.5?
Which term of the G.P. :
\[\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, \frac{1}{4\sqrt{2}}, . . . \text { is }\frac{1}{512\sqrt{2}}?\]
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.
The sum of first three terms of a G.P. is 13/12 and their product is − 1. Find the G.P.
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 progression:
1, 3, 9, 27, ... to 8 terms;
Find the sum of the following geometric progression:
4, 2, 1, 1/2 ... to 10 terms.
Find the sum of the following geometric series:
(x +y) + (x2 + xy + y2) + (x3 + x2y + xy2 + y3) + ... to n terms;
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.
If S1, S2, S3 be respectively the sums of n, 2n, 3n terms of a G.P., then prove that \[S_1^2 + S_2^2\] = S1 (S2 + S3).
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.
One side of an equilateral triangle is 18 cm. The mid-points of its sides are joined to form another triangle whose mid-points, in turn, are joined to form still another triangle. The process is continued indefinitely. Find the sum of the (i) perimeters of all the triangles. (ii) areas of all triangles.
The sum of three numbers a, b, c in A.P. is 18. If a and b are each increased by 4 and c is increased by 36, the new numbers form a G.P. Find a, b, c.
If (a − b), (b − c), (c − a) are in G.P., then prove that (a + b + c)2 = 3 (ab + bc + ca)
If xa = xb/2 zb/2 = zc, then prove that \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P.
Find the geometric means of the following pairs of number:
−8 and −2
If in an infinite G.P., first term is equal to 10 times the sum of all successive terms, then its common ratio is
The value of 91/3 . 91/9 . 91/27 ... upto inf, is
Which term of the G.P. 5, 25, 125, 625, … is 510?
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?
A ball is dropped from a height of 80 ft. The ball is such that it rebounds `(3/4)^"th"` of the height it has fallen. How high does the ball rebound on 6th bounce? How high does the ball rebound on nth bounce?
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 3 years.
The numbers x − 6, 2x and x2 are in G.P. Find x
Find: `sum_("r" = 1)^10(3 xx 2^"r")`
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:
For a sequence Sn = 4(7n – 1) verify that the sequence is a G.P.
Answer the following:
Find k so that k – 1, k, k + 2 are consecutive terms of a G.P.
Answer the following:
If for a G.P. first term is (27)2 and seventh term is (8)2, find S8
At the end of each year the value of a certain machine has depreciated by 20% of its value at the beginning of that year. If its initial value was Rs 1250, find the value at the end of 5 years.
If a, b, c, d are four distinct positive quantities in G.P., then show that a + d > b + c
If pth, qth, and rth terms of an A.P. and G.P. are both a, b and c respectively, show that ab–c . bc – a . ca – b = 1
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 ______.
The sum of infinite number of terms of a decreasing G.P. is 4 and the sum of the terms to m squares of its terms to infinity is `16/3`, then the G.P. is ______.
