Advertisements
Advertisements
प्रश्न
If Sn = n2 p and Sm = m2 p, m ≠ n, in an A.P., prove that Sp = p3.
Advertisements
उत्तर
\[S_n = n^2 p\]
\[ \Rightarrow \frac{n}{2}\left[ 2a + (n - 1)d \right] = n^2 p\]
\[ \Rightarrow 2np = 2a + (n - 1)d . . . (i)\]
\[ S_m = m^2 p\]
\[ \Rightarrow \frac{m}{2}\left[ 2a + (m - 1)d \right] = m^2 p\]
\[ \Rightarrow 2mp = 2a + (m - 1)d . . . (ii)\]
\[\text { Subtracting (ii) from (i), we get }: \]
\[2p(n - m) = (n - m)d\]
\[ \Rightarrow 2p = d . . . (iii)\]
\[\text { Substituing the value in (i), we get }: \]
\[nd = 2a + (n - 1)d\]
\[ \Rightarrow nd - nd + d = 2a\]
\[ \Rightarrow a = \frac{d}{2} = p \left[ \text { from }(iii) \right] . . . (iv)\]
\[ \therefore S_p = \frac{p}{2}\left[ 2a + \left( p - 1 \right)d \right]\]
\[ \Rightarrow S_p = \frac{p}{2}\left[ 2p + \left( p - 1 \right)2p \right]\]
\[ \Rightarrow S_p = \frac{p}{2}\left[ 2p + 2 p^2 - 2p \right]\]
\[ \Rightarrow S_p = \frac{p}{2}\left[ 2 p^2 \right]\]
\[ \Rightarrow S_p = p^3 \]
APPEARS IN
संबंधित प्रश्न
If the sum of a certain number of terms of the A.P. 25, 22, 19, … is 116. Find the last term
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`
A man starts repaying a loan as first installment of Rs. 100. If he increases the installment by Rs 5 every month, what amount he will pay in the 30th installment?
Let the sum of n, 2n, 3n terms of an A.P. be S1, S2 and S3, respectively, show that S3 = 3 (S2– S1)
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.
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
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 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.
The Fibonacci sequence is defined by a1 = 1 = a2, an = an − 1 + an − 2 for n > 2
Find `(""^an +1)/(""^an")` for n = 1, 2, 3, 4, 5.
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.
9, 7, 5, 3, ...
Is 68 a term of the A.P. 7, 10, 13, ...?
In a certain A.P. the 24th term is twice the 10th term. Prove that the 72nd term is twice the 34th term.
How many numbers of two digit are divisible by 3?
The sum of three terms of an A.P. is 21 and the product of the first and the third terms exceeds the second term by 6, find three terms.
If the sum of three numbers in A.P. is 24 and their product is 440, find the numbers.
Find the sum of the following arithmetic progression :
1, 3, 5, 7, ... to 12 terms
Find the sum of the following arithmetic progression :
41, 36, 31, ... to 12 terms
Find the sum of the following arithmetic progression :
(x − y)2, (x2 + y2), (x + y)2, ... to n terms
Find the sum of all integers between 84 and 719, which are multiples of 5.
Solve:
25 + 22 + 19 + 16 + ... + x = 115
If 12th term of an A.P. is −13 and the sum of the first four terms is 24, what is the sum of first 10 terms?
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 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?
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 the sum of p terms of an A.P. is q and the sum of q terms is p, then the sum of p + q terms will be
If the sum of n terms of an A.P., is 3 n2 + 5 n then which of its terms is 164?
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 =
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:
The 10th common term between the A.P.s 3, 7, 11, 15, ... and 1, 6, 11, 16, ... is
Find the sum of first 24 terms of the A.P. a1, a2, a3, ... if it is known that a1 + a5 + a10 + a15 + a20 + a24 = 225.
A man accepts a position with an initial salary of Rs 5200 per month. It is understood that he will receive an automatic increase of Rs 320 in the very next month and each month thereafter. Find his salary for the tenth month
A man accepts a position with an initial salary of Rs 5200 per month. It is understood that he will receive an automatic increase of Rs 320 in the very next month and each month thereafter. What is his total earnings during the first year?
Let Sn denote the sum of the first n terms of an A.P. If S2n = 3Sn then S3n: Sn is equal to ______.
Any term of an A.P. (except first) is equal to half the sum of terms which are equidistant from it.
Let 3, 6, 9, 12 ....... upto 78 terms and 5, 9, 13, 17 ...... upto 59 be two series. Then, the sum of the terms common to both the series is equal to ______.
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 ______.
