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
The sums of n terms of two arithmetic progressions are in the ratio 5n + 4: 9n + 6. Find the ratio of their 18th terms
Find the sum of all two digit numbers which when divided by 4, yields 1 as remainder.
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?
Shamshad Ali buys a scooter for Rs 22000. He pays Rs 4000 cash and agrees to pay the balance in annual installment of Rs 1000 plus 10% interest on the unpaid amount. How much will the scooter cost him?
A sequence is defined by an = n3 − 6n2 + 11n − 6, n ϵ N. Show that the first three terms of the sequence are zero and all other terms are positive.
Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.
3, −1, −5, −9 ...
If the sequence < an > is an A.P., show that am +n +am − n = 2am.
Is 302 a term of the A.P. 3, 8, 13, ...?
If 9th term of an A.P. is zero, prove that its 29th term is double the 19th term.
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 are there between 1 and 1000 which when divided by 7 leave remainder 4?
The first and the last terms of an A.P. are a and l respectively. Show that the sum of nthterm from the beginning and nth term from the end is a + l.
\[\text { If } \theta_1 , \theta_2 , \theta_3 , . . . , \theta_n \text { are in AP, whose common difference is d, then show that }\]
\[\sec \theta_1 \sec \theta_2 + \sec \theta_2 \sec \theta_3 + . . . + \sec \theta_{n - 1} \sec \theta_n = \frac{\tan \theta_n - \tan \theta_1}{\sin d} \left[ NCERT \hspace{0.167em} EXEMPLAR \right]\]
Find the four numbers in A.P., whose sum is 50 and in which the greatest number is 4 times the least.
The angles of a quadrilateral are in A.P. whose common difference is 10°. Find the angles.
Find the sum of the following arithmetic progression :
50, 46, 42, ... to 10 terms
Find the sum of the following serie:
2 + 5 + 8 + ... + 182
Find the r th term of an A.P., the sum of whose first n terms is 3n2 + 2n.
Find the sum of n terms of the A.P. whose kth terms is 5k + 1.
How many terms of the A.P. −6, \[- \frac{11}{2}\], −5, ... are needed to give the sum −25?
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.
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 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 a, b, c is in A.P., then show that:
bc − a2, ca − b2, ab − c2 are in A.P.
If a, b, c is in A.P., prove that:
a2 + c2 + 4ac = 2 (ab + bc + ca)
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.
If x, y, z are in A.P. and A1 is the A.M. of x and y and A2 is the A.M. of y and z, then prove that the A.M. of A1 and A2 is y.
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.
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?
A man saved ₹66000 in 20 years. In each succeeding year after the first year he saved ₹200 more than what he saved in the previous year. How much did he save in the first year?
If 7th and 13th terms of an A.P. be 34 and 64 respectively, then its 18th term is
If the sum of n terms of an A.P. is 2 n2 + 5 n, then its nth 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
The product of three numbers in A.P. is 224, and the largest number is 7 times the smallest. Find the numbers
Any term of an A.P. (except first) is equal to half the sum of terms which are equidistant from it.
If the sum of n terms of a sequence is quadratic expression then it always represents an A.P
