Advertisements
Advertisements
Question
If the sum of first 7 terms of an A.P. is 49 and that of its first 17 terms is 289, find the sum of first n terms of the A.P.
Advertisements
Solution
Let the first term and the common difference of the given A.P. be a and d, respectively.
Sum of the first 7 terms, S7 = 49
We know
`S = n/2[2a + (n - 1)d]`
⇒ `7/2(2a + 6d) = 49`
⇒ `7/2 xx 2(a + 3d) = 49`
⇒ a + 3d = 7 ...(1)
Sum of the first 17 terms, S17 = 289
⇒ `17/2(2a + 16d) = 289`
⇒ `17/2 xx 2(a + 8d) = 289`
⇒ a + 8d = `289/17`
⇒ a + 8d = 17 ...(2)
Subtracting (2) from (1), we get
5d = 10
d = `5/10`
⇒ d = 2
Substituting the value of d in (1), we get
a = 1
Now,
sum of the first n terms is given by
`S_n = n/2[2a + (n - 1)d]`
= `n/2[2 xx 1 + (n - 1) xx 2]`
= `n/2 [2 + 2n - 2]`
= `n/2 [2n]`
= n2
Therefore, the sum of the first n terms of the A.P. is n2.
APPEARS IN
RELATED QUESTIONS
The sum of the first p, q, r terms of an A.P. are a, b, c respectively. Show that `\frac { a }{ p } (q – r) + \frac { b }{ q } (r – p) + \frac { c }{ r } (p – q) = 0`
Check whether -150 is a term of the A.P. 11, 8, 5, 2, ....
Ramkali saved Rs 5 in the first week of a year and then increased her weekly saving by Rs 1.75. If in the nth week, her week, her weekly savings become Rs 20.75, find n.
In an AP, given a = 2, d = 8, and Sn = 90, find n and an.
The first term of an A.P. is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.
A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A of radii 0.5, 1.0 cm, 1.5 cm, 2.0 cm, .... as shown in figure. What is the total length of such a spiral made up of thirteen consecutive semicircles? (Take `pi = 22/7`)
[Hint: Length of successive semicircles is l1, l2, l3, l4, ... with centres at A, B, A, B, ... respectively.]
Find the sum of first 22 terms of an A.P. in which d = 22 and a = 149.
Find the sum of the first 11 terms of the A.P : 2, 6, 10, 14, ...
Find the sum of all 3-digit natural numbers, which are multiples of 11.
In a flower bed, there are 43 rose plants in the first row, 41 in second, 39 in the third, and so on. There are 11 rose plants in the last row. How many rows are there in the flower bed?
If k,(2k - 1) and (2k - 1) are the three successive terms of an AP, find the value of k.
The first term of an AP is p and its common difference is q. Find its 10th term.
Find an AP whose 4th term is 9 and the sum of its 6th and 13th terms is 40.
Find the sum of the following Aps:
i) 2, 7, 12, 17, ……. to 19 terms .
How many terms of the AP 63, 60, 57, 54, ….. must be taken so that their sum is 693? Explain the double answer.
Write an A.P. whose first term is a and common difference is d in the following.
a = –19, d = –4
Choose the correct alternative answer for the following question.
For an given A.P. a = 3.5, d = 0, n = 101, then tn = ....
If the common differences of an A.P. is 3, then a20 − a15 is
There are 25 rows of seats in an auditorium. The first row is of 20 seats, the second of 22 seats, the third of 24 seats, and so on. How many chairs are there in the 21st row ?
Find the sum of n terms of the series \[\left( 4 - \frac{1}{n} \right) + \left( 4 - \frac{2}{n} \right) + \left( 4 - \frac{3}{n} \right) + . . . . . . . . . .\]
The first term of an A.P. is p and its common difference is q. Find its 10th term.
If the sum of first n even natural numbers is equal to k times the sum of first n odd natural numbers, then k =
If 18, a, b, −3 are in A.P., the a + b =
Q.19
Find the sum of first 10 terms of the A.P.
4 + 6 + 8 + .............
If the nth term of an AP is (2n +1), then the sum of its first three terms is ______.
How many terms of the AP: –15, –13, –11,... are needed to make the sum –55? Explain the reason for double answer.
Sum of 1 to n natural number is 45, then find the value of n.
The 5th term and the 9th term of an Arithmetic Progression are 4 and – 12 respectively.
Find:
- the first term
- common difference
- sum of 16 terms of the AP.
