Advertisements
Advertisements
Question
If the sum of first p terms of an AP is 2 (ap2 + bp), find its common difference.
Advertisements
Solution
Let Sp denotes the sum of first p terms of the AP.
∴ sp = ap2 + bp
⇒ `s_(p-1) = a (p-1)^2 + b( p-1)`
= a( p2 - 2p +1 ) +b (p-1)
= ap2 - ( 2a -b) p+ (a-b)
Now,
pth term of AP ` a_p = s_p - s_(p-1)`
= `(ap^2 + bp ) - [ ap^2 -( 2a-b) p+ (a-b) ]`
= `ap^2 + bp - ap^2 + (2a - b ) p-(a-b)`
= 2ap - (a-b)
Let d be the common difference of the AP.
∴ d = `a_p - a_( p-1)`
= [ 2 ap - (a-b) ] = [ 2a (p-1) - (a-b) ]
= 2ap - (a-b) - 2a (p-1 ) + (a-b)
= 2a
Hence, the common difference of the AP is 2a.
APPEARS IN
RELATED QUESTIONS
The houses in a row numbered consecutively from 1 to 49. Show that there exists a value of x such that sum of numbers of houses preceding the house numbered x is equal to sum of the numbers of houses following x.
Find the sum of first n odd natural numbers
If 10 times the 10th term of an AP is equal to 15 times the 15th term, show that its 25th term is zero.
Which term of the AP 21, 18, 15, … is zero?
If `4/5 `, a, 2 are in AP, find the value of a.
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.
The next term of the A.P. \[\sqrt{7}, \sqrt{28}, \sqrt{63}\] is ______.
Write the nth term of an A.P. the sum of whose n terms is Sn.
For what value of p are 2p + 1, 13, 5p − 3 are three consecutive terms of an A.P.?
If k, 2k − 1 and 2k + 1 are three consecutive terms of an A.P., the value of k is
A sum of Rs. 700 is to be paid to give seven cash prizes to the students of a school for their overall academic performance. If the cost of each prize is Rs. 20 less than its preceding prize; find the value of each of the prizes.
Q.5
If the second term and the fourth term of an A.P. are 12 and 20 respectively, then find the sum of first 25 terms:
Find the sum of first 1000 positive integers.
Activity :- Let 1 + 2 + 3 + ........ + 1000
Using formula for the sum of first n terms of an A.P.,
Sn = `square`
S1000 = `square/2 (1 + 1000)`
= 500 × 1001
= `square`
Therefore, Sum of the first 1000 positive integer is `square`
Find t21, if S41 = 4510 in an A.P.
If the nth term of an AP is (2n +1), then the sum of its first three terms is ______.
The first term of an AP of consecutive integers is p2 + 1. The sum of 2p + 1 terms of this AP is ______.
Find the sum of first 16 terms of the A.P. whose nth term is given by an = 5n – 3.
The nth term of an A.P. is 6n + 4. The sum of its first 2 terms is ______.
