Advertisements
Advertisements
प्रश्न
Find the sum of the following geometric series:
1, −a, a2, −a3, ....to n terms (a ≠ 1)
Advertisements
उत्तर
Common Ratio = r = `(-a)/1 = -a`
∴ Sum of GP for n terms = `(a(r^n - 1))/(r - 1)` ...(i)
⇒ a = 1, r = −a, n = n
∴ Substituting the above values in (1) we get
⇒ `(1((-a)^n - 1))/(-a-1)`
⇒ `(-1((-a)^n - 1))/(a+1)`
संबंधित प्रश्न
Find the sum to indicated number of terms in the geometric progressions x3, x5, x7, ... n terms (if x ≠ ± 1).
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 and d are in G.P. show that (a2 + b2 + c2) (b2 + c2 + d2) = (ab + bc + cd)2 .
The sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio `(3 + 2sqrt2) ":" (3 - 2sqrt2)`.
The sum of some terms of G.P. is 315 whose first term and the common ratio are 5 and 2, respectively. Find the last term and the number of terms.
If a, b, c, d are in G.P, prove that (an + bn), (bn + cn), (cn + dn) are in G.P.
Which term of the progression 18, −12, 8, ... is \[\frac{512}{729}\] ?
The seventh term of a G.P. is 8 times the fourth term and 5th term is 48. Find the G.P.
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 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.
Find the sum of the following geometric progression:
(a2 − b2), (a − b), \[\left( \frac{a - b}{a + b} \right)\] to n terms;
Find the sum of the following geometric series:
\[\frac{2}{9} - \frac{1}{3} + \frac{1}{2} - \frac{3}{4} + . . . \text { to 5 terms };\]
Find the sum of the following geometric series:
`3/5 + 4/5^2 + 3/5^3 + 4/5^4 + ....` to 2n terms;
Find the sum of the following series:
0.5 + 0.55 + 0.555 + ... 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.
Find the sum of the following serie to infinity:
8 + \[4\sqrt{2}\] + 4 + ... ∞
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 the rational numbers having the following decimal expansion:
\[0 . \overline3\]
If a, b, c, d are in G.P., prove that:
(a + b + c + d)2 = (a + b)2 + 2 (b + c)2 + (c + d)2
Insert 5 geometric means between 16 and \[\frac{1}{4}\] .
Find the geometric means of the following pairs of number:
2 and 8
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
The numbers x − 6, 2x and x2 are in G.P. Find nth term
For the following G.P.s, find Sn
3, 6, 12, 24, ...
For the following G.P.s, find Sn.
p, q, `"q"^2/"p", "q"^3/"p"^2,` ...
For a G.P. if S5 = 1023 , r = 4, Find a
For a G.P. if a = 2, r = 3, Sn = 242 find n
Find the sum to n terms of the sequence.
0.2, 0.02, 0.002, ...
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.
The midpoints of the sides of a square of side 1 are joined to form a new square. This procedure is repeated indefinitely. Find the sum of the perimeters of all the squares
Select the correct answer from the given alternative.
If common ratio of the G.P is 5, 5th term is 1875, the first term is -
Answer the following:
For a G.P. a = `4/3` and t7 = `243/1024`, find the value of r
Answer the following:
If for a G.P. t3 = `1/3`, t6 = `1/81` find r
Answer the following:
If p, q, r, s are in G.P., show that (p2 + q2 + r2) (q2 + r2 + s2) = (pq + qr + rs)2
If a, b, c, d are four distinct positive quantities in G.P., then show that a + d > b + c
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 ______.
