Advertisements
Advertisements
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.
Advertisements
उत्तर
Let a and d be the first term and the common difference of the A.P. respectively.
It is known that the kth term of an A. P. is given by
ak = a + (k –1) d
∴ am + n = a + (m + n –1) d
am – n = a + (m – n –1) d
am = a + (m –1) d
∴ am + n + am – n = a + (m + n –1) d + a + (m – n –1) d
= 2a + (m + n –1 + m – n –1) d
= 2a + (2m – 2) d
= 2a + 2 (m – 1) d
=2 [a + (m – 1) d]
= 2am
Thus, the sum of (m + n)th and (m – n)th terms of an A.P. is equal to twice the mth term.
APPEARS IN
संबंधित प्रश्न
Find the sum to n terms of the A.P., whose kth term is 5k + 1.
Between 1 and 31, m numbers have been inserted in such a way that the resulting sequence is an A.P. and the ratio of 7th and (m – 1)th numbers is 5:9. Find the value of m.
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
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.
Let < an > be a sequence. Write the first five term in the following:
a1 = a2 = 2, an = an − 1 − 1, n > 2
Which term of the A.P. 3, 8, 13, ... is 248?
Is 302 a term of the A.P. 3, 8, 13, ...?
Which term of the sequence 12 + 8i, 11 + 6i, 10 + 4i, ... is purely real ?
Which term of the sequence 12 + 8i, 11 + 6i, 10 + 4i, ... is purely imaginary?
If the nth term of the A.P. 9, 7, 5, ... is same as the nth term of the A.P. 15, 12, 9, ... find n.
Find the second term and nth term of an A.P. whose 6th term is 12 and the 8th term is 22.
How many numbers are there between 1 and 1000 which when divided by 7 leave remainder 4?
\[\text { If } \theta_1 , \theta_2 , \theta_3 , . . . , \theta_n \text { are in AP, whose common difference is d, then show that }\]
\[\sec \theta_1 \sec \theta_2 + \sec \theta_2 \sec \theta_3 + . . . + \sec \theta_{n - 1} \sec \theta_n = \frac{\tan \theta_n - \tan \theta_1}{\sin d} \left[ NCERT \hspace{0.167em} EXEMPLAR \right]\]
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.
Find the four numbers in A.P., whose sum is 50 and in which the greatest number is 4 times the least.
Find the sum of the following arithmetic progression :
50, 46, 42, ... to 10 terms
Find the sum of the following arithmetic progression :
3, 9/2, 6, 15/2, ... to 25 terms
Find the sum of all integers between 50 and 500 which are divisible by 7.
Solve:
25 + 22 + 19 + 16 + ... + x = 115
If a, b, c is in A.P., then show that:
bc − a2, ca − b2, ab − c2 are 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, b, c is in A.P., prove that:
a2 + c2 + 4ac = 2 (ab + bc + ca)
A piece of equipment cost a certain factory Rs 600,000. If it depreciates in value, 15% the first, 13.5% the next year, 12% the third year, and so on. What will be its value at the end of 10 years, all percentages applying to the original cost?
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.
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?
In a cricket team tournament 16 teams participated. A sum of ₹8000 is to be awarded among themselves as prize money. If the last place team is awarded ₹275 in prize money and the award increases by the same amount for successive finishing places, then how much amount will the first place team receive?
If the sums of n terms of two arithmetic progressions are in the ratio 2n + 5 : 3n + 4, then write the ratio of their m th 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
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 =
If in an A.P., Sn = n2p and Sm = m2p, where Sr denotes the sum of r terms of the A.P., then Sp is equal to
If a1, a2, ..., an are in A.P. with common difference d (where d ≠ 0); then the sum of the series sin d (cosec a1 cosec a2 + cosec a2 cosec a3 + ...+ cosec an–1 cosec an) is equal to cot a1 – cot an
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 the ratio of the sum of n terms of two APs is 2n:(n + 1), then the ratio of their 8th terms is ______.
