English

Let the Sum of N, 2n, 3n Terms of an Arithmetic Progression. Be S1, S2 and S3, Respectively, Show that S3 = 3 (S2– S1) - Mathematics

Advertisements
Advertisements

Question

Let the sum of n, 2n, 3n terms of an A.P. be S1, S2 and S3, respectively, show that S3 = 3 (S2– S1)

Advertisements

Solution

Let a and b be the first term and the common difference of the A.P. respectively.

Therefore,

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

APPEARS IN

NCERT Mathematics [English] Class 11
Chapter 9 Sequences and Series
Miscellaneous Exercise | Q 3 | Page 199

Video TutorialsVIEW ALL [1]

RELATED QUESTIONS

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


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


If the sum of first p terms of an A.P. is equal to the sum of the first q terms, then find the sum of the first (p + q) terms.


If the sum of n terms of an A.P. is 3n2 + 5n and its mth term is 164, find the value of m.


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.


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.


A man deposited Rs 10000 in a bank at the rate of 5% simple interest annually. Find the amount in 15th year since he deposited the amount and also calculate the total amount after 20 years.


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.

−1, 1/4, 3/2, 11/4, ...


The nth term of a sequence is given by an = 2n2 + n + 1. Show that it is not an A.P.


If the sequence < an > is an A.P., show that am +n +am − n = 2am.


Is 68 a term of the A.P. 7, 10, 13, ...?


How many terms are there in the A.P. 7, 10, 13, ... 43 ?


How many terms are there in the A.P.\[- 1, - \frac{5}{6}, -\frac{2}{3}, - \frac{1}{2}, . . . , \frac{10}{3}?\] 


The first term of an A.P. is 5, the common difference is 3 and the last term is 80; find the number of terms.


Find the 12th term from the following arithmetic progression:

3, 8, 13, ..., 253


How many numbers of two digit are divisible by 3?


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


If a2, b2, c2 are in A.P., prove that \[\frac{a}{b + c}, \frac{b}{c + a}, \frac{c}{a + b}\] are 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:

 a3 + c3 + 6abc = 8b3.


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 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.


There are 25 trees at equal distances of 5 metres in a line with a well, the distance of the well from the nearest tree being 10 metres. A gardener waters all the trees separately starting from the well and he returns to the well after watering each tree to get water for the next. Find the total distance the gardener will cover in order to water all the trees.


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?


Write the common difference of an A.P. whose nth term is xn + y.


Write the value of n for which n th terms of the A.P.s 3, 10, 17, ... and 63, 65, 67, .... are equal.


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?


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


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


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


The first term of an A.P. is a, the second term is b and the last term is c. Show that the sum of the A.P. is `((b + c - 2a)(c + a))/(2(b - a))`.


Show that (x2 + xy + y2), (z2 + xz + x2) and (y2 + yz + z2) are consecutive terms of an A.P., if x, y and z are in A.P.


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


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 ______.


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


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×