Advertisements
Advertisements
प्रश्न
Find the sum of 2n terms of the series whose every even term is 'a' times the term before it and every odd term is 'c' times the term before it, the first term being unity.
Advertisements
उत्तर
\[\text { Let the given series be }a_1 + a_2 + a_3 + a_4 + . . . + a_{2n} . \]
\[\text { Now, it is given that }a_1 = 1, a_2 = a a_1 , a_3 = c a_2 , a_4 = a a_3 , a_5 = c a_4 \text { and so on } . \]
\[ \because a_1 = 1\]
\[ \Rightarrow a_1 = 1, a_2 = a, a_3 = ac, a_4 = a^2 c, a_5 = a^2 c^{2,} a_6 = a^3 c^2 , . . . . . \]
\[ \therefore\text { Sum of the 2n terms of the series }, \]
\[ S_n = a_1 + a_2 + a_3 + a_4 + . . . + a_{2n} \]
\[ = 1 + a + ac + a^2 c + a^2 c^2 + . . . + 2n \text { terms }\]
\[ = \left( 1 + a \right) + ac\left( 1 + a \right) + a^2 c^2 \left( 1 + a \right) + . . . + \text { n terms }\]
\[ = \left( 1 + a \right)\left\{ \frac{1 - \left( ac \right)^n}{1 - ac} \right\} \]
\[ = \left( 1 + a \right) \left\{ \frac{\left( ac \right)^n - 1}{ac - 1} \right\}\]
APPEARS IN
संबंधित प्रश्न
Which term of the following sequence:
`sqrt3, 3, 3sqrt3`, .... is 729?
Find the sum to indicated number of terms in the geometric progressions 1, – a, a2, – a3, ... n terms (if a ≠ – 1).
Find the sum to indicated number of terms in the geometric progressions x3, x5, x7, ... n terms (if x ≠ ± 1).
How many terms of G.P. 3, 32, 33, … are needed to give the sum 120?
Find a G.P. for which sum of the first two terms is –4 and the fifth term is 4 times the third term.
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`.
Which term of the G.P. :
\[\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, \frac{1}{4\sqrt{2}}, . . . \text { is }\frac{1}{512\sqrt{2}}?\]
Find the 4th term from the end of the G.P.
\[\frac{1}{2}, \frac{1}{6}, \frac{1}{18}, \frac{1}{54}, . . . , \frac{1}{4374}\]
The seventh term of a G.P. is 8 times the fourth term and 5th term is 48. Find the G.P.
If the G.P.'s 5, 10, 20, ... and 1280, 640, 320, ... have their nth terms equal, find the value of n.
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 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:
2, 6, 18, ... to 7 terms;
Find the sum of the following geometric series:
0.15 + 0.015 + 0.0015 + ... to 8 terms;
Find the sum of the following geometric series:
x3, x5, x7, ... to n terms
Find the sum of the following series:
0.5 + 0.55 + 0.555 + ... to n terms.
The sum of n terms of the G.P. 3, 6, 12, ... is 381. Find the value of n.
The fifth term of a G.P. is 81 whereas its second term is 24. Find the series and sum of its first eight terms.
If S1, S2, ..., Sn are the sums of n terms of n G.P.'s whose first term is 1 in each and common ratios are 1, 2, 3, ..., n respectively, then prove that S1 + S2 + 2S3 + 3S4 + ... (n − 1) Sn = 1n + 2n + 3n + ... + nn.
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.
If the 4th, 10th and 16th terms of a G.P. are x, y and z respectively. Prove that x, y, z are in G.P.
If \[\frac{1}{a + b}, \frac{1}{2b}, \frac{1}{b + c}\] are three consecutive terms of an A.P., prove that a, b, c are the three consecutive terms of a G.P.
If a, b, c are three distinct real numbers in G.P. and a + b + c = xb, then prove that either x< −1 or x > 3.
Insert 6 geometric means between 27 and \[\frac{1}{81}\] .
If (p + q)th and (p − q)th terms of a G.P. are m and n respectively, then write is pth term.
If the first term of a G.P. a1, a2, a3, ... is unity such that 4 a2 + 5 a3 is least, then the common ratio of G.P. is
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 for a sequence, tn = `(5^("n"-3))/(2^("n"-3))`, show that the sequence is a G.P. Find its first term and the common ratio
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?
Find : `sum_("r" = 1)^oo (-1/3)^"r"`
Answer the following:
If for a G.P. t3 = `1/3`, t6 = `1/81` find r
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 x, 2y, 3z are in A.P., where the distinct numbers x, y, z are in G.P. then the common ratio of the G.P. is ______.
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 ______.
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 ______.
