Advertisements
Advertisements
Question
If the sum of a certain number of terms of the AP 25, 22, 19, ... is 116. Find the last term.
Advertisements
Solution
The given A.P. is 25,.22,.19.....
Here, a = 25, d = \[22 - 25 = - 3\]
\[S_n = 116\]
\[ \Rightarrow \frac{n}{2}\left[ 2a + \left( n - 1 \right)d \right] = 116\]
\[ \Rightarrow n\left[ 2 \times 25 + \left( n - 1 \right)\left( - 3 \right) \right] = 232\]
\[ \Rightarrow 50n - 3 n^2 + 3n = 232\]
\[ \Rightarrow 3 n^2 - 53n + 232 = 0\]
\[\Rightarrow 3 n^2 - 29n - 24n + 232 = 0\]
\[ \Rightarrow n(3n - 29) - 8(3n - 29) = 0\]
\[ \Rightarrow (3n - 29)(n - 8) = 0\]
\[ \Rightarrow n = \frac{29}{3} \text { or }8\]
\[\text { Since n cannot be a fraction, } n = 8 . \]
\[\text {Thus, the last term }: \]
\[ a_n = a + (n - 1)d\]
\[ \Rightarrow a_8 = 25 + \left( 8 - 1 \right) \times \left( - 3 \right)\]
\[ \Rightarrow a_8 = 4\]
RELATED QUESTIONS
If the sum of n terms of an A.P. is (pn + qn2), where p and q are constants, find the common difference.
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.
Find the sum of all two digit numbers which when divided by 4, yields 1 as remainder.
A manufacturer reckons that the value of a machine, which costs him Rs 15625, will depreciate each year by 20%. Find the estimated value at the end of 5 years.
If the nth term an of a sequence is given by an = n2 − n + 1, write down its first five terms.
Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.
−1, 1/4, 3/2, 11/4, ...
Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.
\[\sqrt{2}, 3\sqrt{2}, 5\sqrt{2}, 7\sqrt{2}, . . .\]
The nth term of a sequence is given by an = 2n + 7. Show that it is an A.P. Also, find its 7th term.
Which term of the sequence 24, \[23\frac{1}{4,} 22\frac{1}{2,} 21\frac{3}{4}\]....... is the first negative 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 three numbers in A.P. is 12 and the sum of their cubes is 288. Find the numbers.
Find the sum of the following serie:
2 + 5 + 8 + ... + 182
Find the sum of the following serie:
(a − b)2 + (a2 + b2) + (a + b)2 + ... + [(a + b)2 + 6ab]
Find the sum of all odd numbers between 100 and 200.
Find the sum of all integers between 84 and 719, which are multiples of 5.
Find the sum of all even integers between 101 and 999.
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).
The sums of first n terms of two A.P.'s are in the ratio (7n + 2) : (n + 4). Find the ratio of their 5th terms.
If \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P., prove that:
\[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] are in A.P.
If \[a\left( \frac{1}{b} + \frac{1}{c} \right), b\left( \frac{1}{c} + \frac{1}{a} \right), c\left( \frac{1}{a} + \frac{1}{b} \right)\] are in A.P., prove that a, b, c are in A.P.
A man saved Rs 16500 in ten years. In each year after the first he saved Rs 100 more than he did in the receding year. How much did he save in the first year?
A man accepts a position with an initial salary of ₹5200 per month. It is understood that he will receive an automatic increase of ₹320 in the very next month and each month thereafter.
(i) Find his salary for the tenth month.
(ii) What is his total earnings during the first year?
Write the sum of first n even natural numbers.
If m th term of an A.P. is n and nth term is m, then write its pth term.
If 7th and 13th terms of an A.P. be 34 and 64 respectively, then its 18th term 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 n arithmetic means are inserted between 1 and 31 such that the ratio of the first mean and nth mean is 3 : 29, then the value of n is
Let Sn denote the sum of n terms of an A.P. whose first term is a. If the common difference d is given by d = Sn − k Sn − 1 + Sn − 2 , 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:
The 10th common term between the A.P.s 3, 7, 11, 15, ... and 1, 6, 11, 16, ... 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
If a, b, c are in A.P. and x, y, z are in G.P., then the value of xb − c yc − a za − b is
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))`.
If there are (2n + 1) terms in an A.P., then prove that the ratio of the sum of odd terms and the sum of even terms is (n + 1) : n
The product of three numbers in A.P. is 224, and the largest number is 7 times the smallest. Find the numbers
Show that (x2 + xy + y2), (z2 + xz + x2) and (y2 + yz + z2) are consecutive terms of an A.P., if x, y and z are in A.P.
If the sum of m terms of an A.P. is equal to the sum of either the next n terms or the next p terms, then prove that `(m + n) (1/m - 1/p) = (m + p) (1/m - 1/n)`
Let Sn denote the sum of the first n terms of an A.P. If S2n = 3Sn then S3n: Sn is equal to ______.
If b2, a2, c2 are in A.P., then `1/(a + b), 1/(b + c), 1/(c + a)` will be in ______
