Advertisements
Advertisements
प्रश्न
Find the sum of the following serie:
(a − b)2 + (a2 + b2) + (a + b)2 + ... + [(a + b)2 + 6ab]
उत्तर
(a − b)2 + (a2 + b2) + (a + b)2 + ... + [(a + b)2 + 6ab]
Here, the series is an A.P. where we have the following:
\[a = (a - b )^2 \]
\[d = \left( a^2 + b^2 - (a - b )^2 \right) = 2ab\]
\[ a_n = {[(a + b)}^2 + 6ab]\]
\[ \Rightarrow (a - b )^2 + (n - 1)(2ab) = {[(a + b)}^2 + 6ab] \]
\[ \Rightarrow a^2 + b^2 - 2ab + 2abn - 2ab = [ a^2 + b^2 + 2ab + 6ab]\]
\[ \Rightarrow a^2 + b^2 - 4ab + 2abn = a^2 + b^2 + 8ab\]
\[ \Rightarrow 2abn = 12ab \]
\[ \Rightarrow n = 6\]
\[ S_n = \frac{n}{2}\left[ 2a + (n - 1)d \right]\]
\[ \Rightarrow S_6 = \frac{6}{2}\left[ 2(a - b )^2 + \left( 6 - 1 \right) 2ab \right] \]
\[ = 3\left[ 2( a^2 + b^2 - 2ab) + 10ab \right]\]
\[ = 3\left[ 2 a^2 + 2 b^2 - 4ab + 10ab \right]\]
\[ = 3\left[ 2 a^2 + 2 b^2 + 6ab \right]\]
\[ = 6\left[ a^2 + b^2 + 3ab \right]\]
APPEARS IN
संबंधित प्रश्न
Show that the sum of (m + n)th and (m – n)th terms of an A.P. is equal to twice the mth term.
If the sum of three numbers in A.P., is 24 and their product is 440, find the numbers.
Find the sum of integers from 1 to 100 that are divisible by 2 or 5.
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?
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 ...
Find:
10th term of the A.P. 1, 4, 7, 10, ...
Which term of the A.P. 84, 80, 76, ... is 0?
Which term of the A.P. 4, 9, 14, ... is 254?
How many terms are there in the A.P. 7, 10, 13, ... 43 ?
If 9th term of an A.P. is zero, prove that its 29th term is double the 19th term.
Find the 12th term from the following arithmetic progression:
3, 5, 7, 9, ... 201
Find the 12th term from the following arithmetic progression:
3, 8, 13, ..., 253
Find the sum of all integers between 84 and 719, which are multiples of 5.
Find the sum of all even integers between 101 and 999.
How many terms are there in the A.P. whose first and fifth terms are −14 and 2 respectively and the sum of the terms is 40?
If Sn = n2 p and Sm = m2 p, m ≠ n, in an A.P., prove that Sp = p3.
If the 5th and 12th terms of an A.P. are 30 and 65 respectively, what is the sum of first 20 terms?
Find the sum of all two digit numbers which when divided by 4, yields 1 as remainder.
Find an A.P. in which the sum of any number of terms is always three times the squared number of these terms.
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:
b + c − a, c + a − b, a + b − c are in A.P.
Show that x2 + xy + y2, z2 + zx + x2 and y2 + yz + z2 are consecutive terms of an A.P., if x, y and z are in A.P.
A man saved Rs 16500 in ten years. In each year after the first he saved Rs 100 more than he did in the receding year. How much did he save in the first year?
A man arranges to pay off a debt of Rs 3600 by 40 annual instalments which form an arithmetic series. When 30 of the instalments are paid, he dies leaving one-third of the debt unpaid, find the value of the first instalment.
Shamshad Ali buys a scooter for Rs 22000. He pays Rs 4000 cash and agrees to pay the balance in annual instalments of Rs 1000 plus 10% interest on the unpaid amount. How much the scooter will cost him.
In a potato race 20 potatoes are placed in a line at intervals of 4 meters with the first potato 24 metres from the starting point. A contestant is required to bring the potatoes back to the starting place one at a time. How far would he run in bringing back all the potatoes?
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 sum of n terms of an AP is 2n2 + 3n, then write its nth term.
Write the sum of first n even natural numbers.
If, S1 is the sum of an arithmetic progression of 'n' odd number of terms and S2 the sum of the terms of the series in odd places, then \[\frac{S_1}{S_2}\] =
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
If there are (2n + 1) terms in an A.P., then prove that the ratio of the sum of odd terms and the sum of even terms is (n + 1) : n
If the sum of p terms of an A.P. is q and the sum of q terms is p, show that the sum of p + q terms is – (p + q). Also, find the sum of first p – q terms (p > q).
If in an A.P., Sn = qn2 and Sm = qm2, where Sr denotes the sum of r terms of the A.P., then Sq equals ______.
If a1, a2, a3, .......... are an A.P. such that a1 + a5 + a10 + a15 + a20 + a24 = 225, then a1 + a2 + a3 + ...... + a23 + a24 is equal to ______.
The fourth term of an A.P. is three times of the first term and the seventh term exceeds the twice of the third term by one, then the common difference of the progression is ______.
