Advertisements
Advertisements
Question
Evaluate the following:
\[\sum^n_{k = 1} ( 2^k + 3^{k - 1} )\]
Advertisements
Solution
\[S_n = \sum^n_{k = 1} \left( 2^k + 3^{k - 1} \right)\]
\[ = \sum^n_{k = 1} 2^k + \sum^n_{k = 1} 3^{k - 1} \]
\[ = \left( 2 + 4 + 8 + . . . + 2^n \right) + \left( 1 + 3 + 9 + . . . + 3^n \right) \]
\[ = 2\left( \frac{2^n - 1}{2 - 1} \right) + 1\left( \frac{3^n - 1}{3 - 1} \right) \]
\[ = \frac{1}{2}\left( 2^{n + 2} - 4 + 3^n - 1 \right) \]
\[ = \frac{1}{2}\left( 2^{n + 2} + 3^n - 5 \right)\]
APPEARS IN
RELATED QUESTIONS
How many terms of G.P. 3, 32, 33, … are needed to give the sum 120?
Given a G.P. with a = 729 and 7th term 64, determine S7.
Find the sum to n terms of the sequence, 8, 88, 888, 8888… .
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.
A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of terms occupying odd places, then find its common ratio.
Which term of the G.P.: `sqrt3, 3, 3sqrt3`, ... is 729?
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:
2, 6, 18, ... to 7 terms;
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:
`sqrt7, sqrt21, 3sqrt7,...` to n terms
The ratio of the sum of the first three terms to that of the first 6 terms of a G.P. is 125 : 152. Find the common ratio.
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).
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.
The sum of first two terms of an infinite G.P. is 5 and each term is three times the sum of the succeeding terms. Find the G.P.
If a, b, c, d are in G.P., prove that:
(a2 − b2), (b2 − c2), (c2 − d2) are in G.P.
If (a − b), (b − c), (c − a) are in G.P., then prove that (a + b + c)2 = 3 (ab + bc + ca)
If a, b, c are in A.P. and a, b, d are in G.P., then prove that a, a − b, d − c are in G.P.
Insert 5 geometric means between 16 and \[\frac{1}{4}\] .
The fractional value of 2.357 is
If pth, qth and rth terms of an A.P. are in G.P., then the common ratio of this G.P. is
For the G.P. if a = `2/3`, t6 = 162, find r.
Which term of the G.P. 5, 25, 125, 625, … is 510?
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 fifth term of a G.P. is x, eighth term of a G.P. is y and eleventh term of a G.P. is z verify whether y2 = xz
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 n years.
For a G.P. a = 2, r = `-2/3`, find S6
For a G.P. if a = 2, r = 3, Sn = 242 find n
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 5 xx 3^"r"`
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]
If the A.M. of two numbers exceeds their G.M. by 2 and their H.M. by `18/5`, find the numbers.
Select the correct answer from the given alternative.
The common ratio for the G.P. 0.12, 0.24, 0.48, is –
Answer the following:
Which 2 terms are inserted between 5 and 40 so that the resulting sequence is G.P.
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 ______.
The third term of G.P. is 4. The product of its first 5 terms is ______.
The third term of a G.P. is 4, the product of the first five terms is ______.
Let `{a_n}_(n = 0)^∞` be a sequence such that a0 = a1 = 0 and an+2 = 2an+1 – an + 1 for all n ≥ 0. Then, `sum_(n = 2)^∞ a^n/7^n` is equal to ______.
