Advertisements
Advertisements
Question
The 4th term of an A.P. is three times the first and the 7th term exceeds twice the third term by 1. Find the first term and the common difference.
Advertisements
Solution
Given:
Let the first term of the A.P. be a and the common difference be d.
\[a_4 = 3a\]
\[ \Rightarrow a + \left( 4 - 1 \right)d = 3a\]
\[ \Rightarrow a + 3d = 3a\]
\[ \Rightarrow 3d = 2a\]
\[ \Rightarrow a = \frac{3d}{2} . . . (i)\]
\[\text { And,} a_7 - 2 a_3 = 1\]
\[ \Rightarrow a + \left( 7 - 1 \right)d - 2\left[ a + \left( 3 - 1 \right)d \right] = 1\]
\[ \Rightarrow a + 6d - 2(a + 2d) = 1\]
\[ \Rightarrow a + 6d - 2a - 4d = 1\]
\[ \Rightarrow - a + 2d = 1\]
\[ \Rightarrow - \frac{3d}{2} + 2d = 1 \left[ \text { From } (i) \right] \]
\[ \Rightarrow \frac{- 3d + 4d}{2} = 1\]
\[ \Rightarrow \frac{d}{2} = 1\]
\[ \Rightarrow d = 2\]
\[\text { Putting the value in (i), we get }: \]
\[a = \frac{3 \times 2}{2}\]
\[ \Rightarrow a = 3\]
APPEARS IN
RELATED QUESTIONS
Find the sum of odd integers from 1 to 2001.
If the sum of a certain number of terms of the A.P. 25, 22, 19, … is 116. Find the last term
If the sum of first p terms of an A.P. is equal to the sum of the first q terms, then find the sum of the first (p + q) terms.
if `(a^n + b^n)/(a^(n-1) + b^(n-1))` is the A.M. between a and b, then find the value of n.
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.
A farmer buys a used tractor for Rs 12000. He pays Rs 6000 cash and agrees to pay the balance in annual installments of Rs 500 plus 12% interest on the unpaid amount. How much will be the tractor cost him?
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, ...
How many terms are there in the A.P. 7, 10, 13, ... 43 ?
The 10th and 18th terms of an A.P. are 41 and 73 respectively. Find 26th term.
In a certain A.P. the 24th term is twice the 10th term. Prove that the 72nd term is twice the 34th term.
Find the 12th term from the following arithmetic progression:
3, 5, 7, 9, ... 201
How many numbers are there between 1 and 1000 which when divided by 7 leave remainder 4?
The angles of a quadrilateral are in A.P. whose common difference is 10°. Find the angles.
Find the sum of first n odd natural numbers.
Find the sum of all those integers between 100 and 800 each of which on division by 16 leaves the remainder 7.
The third term of an A.P. is 7 and the seventh term exceeds three times the third term by 2. Find the first term, the common difference and the sum of first 20 terms.
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 a, b, c is in A.P., then show that:
a2 (b + c), b2 (c + a), c2 (a + b) are also 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.
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.
The income of a person is Rs 300,000 in the first year and he receives an increase of Rs 10000 to his income per year for the next 19 years. Find the total amount, he received in 20 years.
A carpenter was hired to build 192 window frames. The first day he made five frames and each day thereafter he made two more frames than he made the day before. How many days did it take him to finish the job?
If m th term of an A.P. is n and nth term is m, then write its pth term.
If the sums of n terms of two AP.'s are in the ratio (3n + 2) : (2n + 3), then find the ratio of their 12th terms.
If 7th and 13th terms of an A.P. be 34 and 64 respectively, then its 18th term is
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
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:
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 G.P. and a1/x = b1/y = c1/z, then xyz are in
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)`
Find the rth term of an A.P. sum of whose first n terms is 2n + 3n2
If the sum of n terms of an A.P. is given by Sn = 3n + 2n2, then the common difference of the A.P. is ______.
If the sum of n terms of a sequence is quadratic expression then it always represents an A.P
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 number of terms in an A.P. is even; the sum of the odd terms in lt is 24 and that the even terms is 30. If the last term exceeds the first term by `10 1/2`, then the number of terms in the A.P. is ______.
