Show that the Sum of (M + N)Th And (M – N)Th Terms of an A.P. is Equal to Twice The Mth Term. - Mathematics

Advertisements
Advertisements

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

Solution

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.

  Is there an error in this question or solution?
Chapter 9: Sequences and Series - Miscellaneous Exercise [Page 199]

APPEARS IN

NCERT Mathematics Class 11
Chapter 9 Sequences and Series
Miscellaneous Exercise | Q 1 | Page 199

Video TutorialsVIEW ALL [1]

RELATED QUESTIONS

If the sum of n terms of an A.P. is (pn qn2), where p and q are constants, find the common difference.


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?


If the nth term an of a sequence is given by an = n2 − n + 1, write down its first five terms.


Let < an > be a sequence. Write the first five term in the following:

a1 = 1, an = an − 1 + 2, n ≥ 2


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.

\[\sqrt{2}, 3\sqrt{2}, 5\sqrt{2}, 7\sqrt{2}, . . .\]


Which term of the sequence 12 + 8i, 11 + 6i, 10 + 4i, ... is purely imaginary?


The 6th and 17th terms of an A.P. are 19 and 41 respectively, find the 40th term.


\[\text { If } \theta_1 , \theta_2 , \theta_3 , . . . , \theta_n \text { are in AP, whose common difference is d, then show that }\]

\[\sec \theta_1 \sec \theta_2 + \sec \theta_2 \sec \theta_3 + . . . + \sec \theta_{n - 1} \sec \theta_n = \frac{\tan \theta_n - \tan \theta_1}{\sin d} \left[ NCERT \hspace{0.167em} EXEMPLAR \right]\]


Find the sum of the following arithmetic progression :

50, 46, 42, ... to 10 terms


Find the sum of the following arithmetic progression :

41, 36, 31, ... to 12 terms


Find the sum of first n natural numbers.


Find the sum of all natural numbers between 1 and 100, which are divisible by 2 or 5.


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.


Find the sum of the series:
3 + 5 + 7 + 6 + 9 + 12 + 9 + 13 + 17 + ... to 3n terms.


Solve: 

1 + 4 + 7 + 10 + ... + x = 590.


The sum of first 7 terms of an A.P. is 10 and that of next 7 terms is 17. Find the progression.


Find the sum of n terms of the A.P. whose kth terms is 5k + 1.


If the sum of a certain number of terms of the AP 25, 22, 19, ... is 116. Find the last term.


If a, b, c is in A.P., then show that:

b + c − a, c + a − b, a + b − c are in A.P.


If \[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] are in A.P., prove that:

\[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P.


A man saves Rs 32 during the first year. Rs 36 in the second year and in this way he increases his savings by Rs 4 every year. Find in what time his saving will be Rs 200.


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.


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?


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 log 2, log (2x − 1) and log (2x + 3) are in A.P., write the value of x.


If \[\frac{3 + 5 + 7 + . . . + \text { upto n terms }}{5 + 8 + 11 + . . . . \text { upto 10 terms }}\] 7, then find the value of n.


If m th term of an A.P. is n and nth term is m, then write its pth term.


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 the sum of n terms of an A.P., is 3 n2 + 5 n then which of its terms is 164?


If a1, a2, a3, .... an are in A.P. with common difference d, then the sum of the series sin d [cosec a1cosec a2 + cosec a1 cosec a3 + .... + cosec an − 1 cosec an] 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


Mark the correct alternative in the following question:

Let Sn denote the sum of first n terms of an A.P. If S2n = 3Sn, then S3n : Sn is equal to


If for an arithmetic progression, 9 times nineth term is equal to 13 times thirteenth term, then value of twenty second term is ____________.


Find the rth term of an A.P. sum of whose first n terms is 2n + 3n2 


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 ______.


If the ratio of the sum of n terms of two APs is 2n:(n + 1), then the ratio of their 8th terms is ______.


If n AM's are inserted between 1 and 31 and ratio of 7th and (n – 1)th A.M. is 5:9, then n equals ______.


The sum of n terms of an AP is 3n2 + 5n. The number of term which equals 164 is ______.


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 ______.


If the first term of an A.P. is 3 and the sum of its first 25 terms is equal to the sum of its next 15 terms, then the common difference of this A.P. is ______.


If 100 times the 100th term of an A.P. with non zero common difference equals the 50 times its 50th term, then the 150th term of this A.P. is ______.


The number of terms in an A.P. is even; the sum of the odd terms in lt is 24 and that the even terms is 30. If the last term exceeds the first term by `10 1/2`, then the number of terms in the A.P. is ______.


If b2, a2, c2 are in A.P., then `1/(a + b), 1/(b + c), 1/(c + a)` will be in ______


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 ______.


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 ______.


Share
Notifications



      Forgot password?
Use app×