Advertisements
Advertisements
Question
In an A.P. the first term is 2 and the sum of the first five terms is one fourth of the next five terms. Show that 20th term is −112.
Advertisements
Solution
\[\text { Given }: \]
\[ a = 2, S_5 = \frac{1}{4}\left( S_{10} - S_5 \right)\]
\[\text { We have: } \]
\[ S_5 = \frac{5}{2} \left[ 2 \times 2 + (5 - 1)d \right]\]
\[ \Rightarrow S_5 = 5\left[ 2 + 2d \right] . . . . (i)\]
\[\text { Also }, S_{10} = \frac{10}{2}\left[ 2 \times 2 + (10 - 1)d \right]\]
\[ \Rightarrow S_{10} = 5\left[ 4 + 9d \right] . . . . . (ii)\]
\[ \because S_5 = \frac{1}{4}\left( S_{10} - S_5 \right) \]
\[\text { From (i) and (ii), we have: } \]
\[ \Rightarrow 5\left[ 2 + 2d \right] = \frac{1}{4}\left[ 5(4 + 9d) - 5(2 + 2d) \right]\]
\[ \Rightarrow 8 + 8d = 4 + 9d - 2 - 2d\]
\[ \Rightarrow d = - 6\]
\[ \therefore a_{20} = a + \left( 20 - 1 \right)d\]
\[ \Rightarrow a_{20} = a + 19d\]
\[ \Rightarrow a_{20} = 2 + 19\left( - 6 \right)\]
\[ \Rightarrow a_{20} = - 112\]
APPEARS IN
RELATED QUESTIONS
The sums of n terms of two arithmetic progressions are in the ratio 5n + 4: 9n + 6. Find the ratio of their 18th terms
Sum of the first p, q and r terms of an A.P. are a, b and c, respectively.
Prove that `a/p (q - r) + b/q (r- p) + c/r (p - q) = 0`
The ratio of the sums of m and n terms of an A.P. is m2: n2. Show that the ratio of mth and nthterm is (2m – 1): (2n – 1)
The difference between any two consecutive interior angles of a polygon is 5°. If the smallest angle is 120°, find the number of the sides of the polygon.
Let the sum of n, 2n, 3n terms of an A.P. be S1, S2 and S3, respectively, show that S3 = 3 (S2– S1)
if `a(1/b + 1/c), b(1/c+1/a), c(1/a+1/b)` are in A.P., prove that a, b, c are in A.P.
Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.
\[\sqrt{2}, 3\sqrt{2}, 5\sqrt{2}, 7\sqrt{2}, . . .\]
Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.
9, 7, 5, 3, ...
Find:
nth term of the A.P. 13, 8, 3, −2, ...
How many terms are there in the A.P.\[- 1, - \frac{5}{6}, -\frac{2}{3}, - \frac{1}{2}, . . . , \frac{10}{3}?\]
The 6th and 17th terms of an A.P. are 19 and 41 respectively, find the 40th term.
If the nth term of the A.P. 9, 7, 5, ... is same as the nth term of the A.P. 15, 12, 9, ... find n.
The sum of 4th and 8th terms of an A.P. is 24 and the sum of the 6th and 10th terms is 34. Find the first term and the common difference of the A.P.
Three numbers are in A.P. If the sum of these numbers be 27 and the product 648, find the numbers.
Find the sum of the following arithmetic progression :
1, 3, 5, 7, ... to 12 terms
Find the sum of all natural numbers between 1 and 100, which are divisible by 2 or 5.
Find the sum of all integers between 50 and 500 which are divisible by 7.
Find the sum of all even integers between 101 and 999.
The first term of an A.P. is 2 and the last term is 50. The sum of all these terms is 442. Find the common difference.
If the sum of a certain number of terms of the AP 25, 22, 19, ... is 116. Find the last term.
If S1 be the sum of (2n + 1) terms of an A.P. and S2 be the sum of its odd terms, then prove that: S1 : S2 = (2n + 1) : (n + 1).
If a, b, c is in A.P., prove that:
a3 + c3 + 6abc = 8b3.
A man saves Rs 32 during the first year. Rs 36 in the second year and in this way he increases his savings by Rs 4 every year. Find in what time his saving will be Rs 200.
A man is employed to count Rs 10710. He counts at the rate of Rs 180 per minute for half an hour. After this he counts at the rate of Rs 3 less every minute than the preceding minute. Find the time taken by him to count the entire amount.
Write the common difference of an A.P. the sum of whose first n terms is
If the sums of n terms of two arithmetic progressions are in the ratio 2n + 5 : 3n + 4, then write the ratio of their m th terms.
Write the value of n for which n th terms of the A.P.s 3, 10, 17, ... and 63, 65, 67, .... are equal.
If a1, a2, a3, .... an are in A.P. with common difference d, then the sum of the series sin d [cosec a1cosec a2 + cosec a1 cosec a3 + .... + cosec an − 1 cosec an] is
In the arithmetic progression whose common difference is non-zero, the sum of first 3 n terms is equal to the sum of next n terms. Then the ratio of the sum of the first 2 n terms to the next 2 nterms is
If four numbers in A.P. are such that their sum is 50 and the greatest number is 4 times the least, then the numbers are
The first and last term of an A.P. are a and l respectively. If S is the sum of all the terms of the A.P. and the common difference is given by \[\frac{l^2 - a^2}{k - (l + a)}\] , then k =
Mark the correct alternative in the following question:
If in an A.P., the pth term is q and (p + q)th term is zero, then the qth term is
Mark the correct alternative in the following question:
Let Sn denote the sum of first n terms of an A.P. If S2n = 3Sn, then S3n : Sn is equal to
The first three of four given numbers are in G.P. and their last three are in A.P. with common difference 6. If first and fourth numbers are equal, then the first number is
If a, b, c are in G.P. and a1/x = b1/y = c1/z, then xyz are in
The first term of an A.P. is a, the second term is b and the last term is c. Show that the sum of the A.P. is `((b + c - 2a)(c + a))/(2(b - a))`.
The first term of an A.P.is a, and the sum of the first p terms is zero, show that the sum of its next q terms is `(-a(p + q)q)/(p - 1)`
Any term of an A.P. (except first) is equal to half the sum of terms which are equidistant from it.
