Advertisements
Advertisements
Question
How many terms of the A.P. −6, \[- \frac{11}{2}\], −5, ... are needed to give the sum −25?
Advertisements
Solution
\[\text { Given: } \]
\[\text{ An } \hspace{0.167em} A . P .\text { with a = - 6 and d }= - \frac{11}{2} - \left( - 6 \right) = \frac{1}{2}\]
\[ S_n = - 25\]
\[ \therefore - 25 = \frac{n}{2}\left[ 2 \times \left( - 6 \right) + \left( n - 1 \right)\frac{1}{2} \right]\]
\[ \Rightarrow - 25 = \frac{n}{2}\left[ - 12 + \frac{n}{2} - \frac{1}{2} \right]\]
\[ \Rightarrow - 50 = n\left[ \frac{n}{2} - \frac{25}{2} \right]\]
\[ \Rightarrow - 100 = n\left( n - 25 \right)\]
\[ \Rightarrow n^2 - 25n + 100 = 0\]
\[ \Rightarrow \left( n - 20 \right)\left( n - 5 \right) = 0\]
\[ \Rightarrow n = 20 \text { or } n = 5\]
RELATED QUESTIONS
Find the sum to n terms of the A.P., whose kth term is 5k + 1.
The sums of n terms of two arithmetic progressions are in the ratio 5n + 4: 9n + 6. Find the ratio of their 18th terms
If the sum of n terms of an A.P. is 3n2 + 5n and its mth term is 164, find the value of m.
The sum of the first four terms of an A.P. is 56. The sum of the last four terms is 112. If its first term is 11, then find the number of terms.
Let < an > be a sequence. Write the first five term in the following:
a1 = 1, an = an − 1 + 2, n ≥ 2
Let < an > be a sequence. Write the first five term in the following:
a1 = 1 = a2, an = an − 1 + an − 2, n > 2
Find:
nth term of the A.P. 13, 8, 3, −2, ...
Which term of the A.P. 84, 80, 76,... is 0?
Which term of the sequence 24, \[23\frac{1}{4,} 22\frac{1}{2,} 21\frac{3}{4}\]....... is the first negative term?
Which term of the sequence 12 + 8i, 11 + 6i, 10 + 4i, ... is purely imaginary?
An A.P. consists of 60 terms. If the first and the last terms be 7 and 125 respectively, find 32nd term.
Find the sum of the following arithmetic progression :
50, 46, 42, ... to 10 terms
Find the sum of the following serie:
101 + 99 + 97 + ... + 47
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 r th term of an A.P., the sum of whose first n terms is 3n2 + 2n.
The sum of first 7 terms of an A.P. is 10 and that of next 7 terms is 17. Find the progression.
If the sum of a certain number of terms of the AP 25, 22, 19, ... is 116. Find the last term.
Find the sum of odd integers from 1 to 2001.
Find an A.P. in which the sum of any number of terms is always three times the squared number of these terms.
If the sum of n terms of an A.P. is nP + \[\frac{1}{2}\] n (n − 1) Q, where P and Q are constants, find the common difference.
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 \[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] are in A.P., prove that:
bc, ca, ab are in A.P.
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 manufacturer of radio sets produced 600 units in the third year and 700 units in the seventh year. Assuming that the product increases uniformly by a fixed number every year, find (i) the production in the first year (ii) the total product in 7 years and (iii) the product in the 10th year.
Sum of all two digit numbers which when divided by 4 yield unity as remainder is
If the sum of n terms of an A.P. is 2 n2 + 5 n, then its nth term is
If the sum of first n even natural numbers is equal to k times the sum of first n odd natural numbers, 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
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 for an arithmetic progression, 9 times nineth term is equal to 13 times thirteenth term, then value of twenty second term is ____________.
If a1, a2, ..., an are in A.P. with common difference d (where d ≠ 0); then the sum of the series sin d (cosec a1 cosec a2 + cosec a2 cosec a3 + ...+ cosec an–1 cosec an) is equal to cot a1 – cot an
A man saved Rs 66000 in 20 years. In each succeeding year after the first year he saved Rs 200 more than what he saved in the previous year. How much did he save in the first year?
If the ratio of the sum of n terms of two APs is 2n:(n + 1), then the ratio of their 8th terms is ______.
If n AM's are inserted between 1 and 31 and ratio of 7th and (n – 1)th A.M. is 5:9, then n equals ______.
The sum of n terms of an AP is 3n2 + 5n. The number of term which equals 164 is ______.
The internal angles of a convex polygon are in A.P. The smallest angle is 120° and the common difference is 5°. The number to sides of the polygon is ______.
What is the general form of an A.P. with first term \(a\) and common difference \(d\)?
