Advertisements
Advertisements
Question
The pth, qth and rth terms of an A.P. are a, b, c respectively. Show that (q – r )a + (r – p )b + (p – q )c = 0
Advertisements
Solution
Let t and d be the first term and the common difference of the A.P. respectively.
The nth term of an A.P. is given by, an = t + (n – 1) d
Therefore,
ap = t + (p – 1) d = a … (1)
aq = t + (q – 1)d = b … (2)
ar = t + (r – 1) d = c … (3)
Subtracting equation (2) from (1), we obtain
(p – 1 – q + 1) d = a – b
⇒ (p – q) d = a – b
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
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)
If the sum of n terms of an A.P. is 3n2 + 5n and its mth term is 164, find the value of m.
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.
A person writes a letter to four of his friends. He asks each one of them to copy the letter and mail to four different persons with instruction that they move the chain similarly. Assuming that the chain is not broken and that it costs 50 paise to mail one letter. Find the amount spent on the postage when 8th set of letter is mailed.
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.
Let < an > be a sequence. Write the first five term in the following:
a1 = 1, an = an − 1 + 2, n ≥ 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.
Find:
nth term of the A.P. 13, 8, 3, −2, ...
If the sequence < an > is an A.P., show that am +n +am − n = 2am.
The first term of an A.P. is 5, the common difference is 3 and the last term is 80; find the number of terms.
If 10 times the 10th term of an A.P. is equal to 15 times the 15th term, show that 25th term of the A.P. is zero.
Find the 12th term from the following arithmetic progression:
3, 8, 13, ..., 253
How many numbers of two digit are divisible by 3?
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.
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 arithmetic progression :
(x − y)2, (x2 + y2), (x + y)2, ... to n terms
Find the sum of the following arithmetic progression :
\[\frac{x - y}{x + y}, \frac{3x - 2y}{x + y}, \frac{5x - 3y}{x + y}\], ... to n terms.
Find the sum of all natural numbers between 1 and 100, which are divisible by 2 or 5.
Show that the sum of all odd integers between 1 and 1000 which are divisible by 3 is 83667.
Find the sum of all integers between 50 and 500 which are divisible by 7.
Solve:
25 + 22 + 19 + 16 + ... + x = 115
Solve:
1 + 4 + 7 + 10 + ... + x = 590.
Find the sum of odd integers from 1 to 2001.
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 \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P., prove that:
\[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] are in A.P.
If a, b, c is in A.P., then show that:
b + c − a, c + a − b, a + b − c are in A.P.
A man starts repaying a loan as first instalment of Rs 100 = 00. If he increases the instalments by Rs 5 every month, what amount he will pay in the 30th instalment?
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.
If the sum of n terms of an A.P. be 3 n2 − n and its common difference is 6, then its first term is
If Sn denotes the sum of first n terms of an A.P. < an > such that
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:
\[\text { If in an A . P } . S_n = n^2 q \text { and } S_m = m^2 q, \text { where } S_r \text{ denotes the sum of r terms of the A . P . , then }S_q \text { equals }\]
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.
Find the rth term of an A.P. sum of whose first n terms is 2n + 3n2
If the ratio of the sum of n terms of two APs is 2n:(n + 1), then the ratio of their 8th terms is ______.
Which of the following best defines an Arithmetic Progression (A.P.)?
