Advertisements
Advertisements
Question
Find the common difference of an AP whose first term is 5 and the sum of its first four terms is half the sum of the next four terms.
Advertisements
Solution
Let the common difference of the AP be d.
First term, a = 5
Now,
`a_1 + a_2 +a_3 +a_4 = 1/2 (a_5 +a_6 +a_7 +a_8)` (Given)
`a+(a +d ) +( a+2d ) +(a+3d) = 1/2 [(a+4d)+(a+5d) +(a+6d) +(a+7d)]`
`[ a_n = a+ (n-1) d]`
⇒ 4a + 6d = `1/2` ( 4a + 22d)
⇒ 8a + 12d = 4a =22d
⇒ 22d - 12 d= 8a + 4a
⇒ 10d = 4a
⇒`d= 2/5 a`
⇒ `d= 2/5 xx 5 = 2 ` (a=5)
Hence, the common difference of the AP is 2.
RELATED QUESTIONS
Find the sum of first 22 terms of an AP in which d = 7 and 22nd term is 149.
Find the sum of the following arithmetic progressions:
3, 9/2, 6, 15/2, ... to 25 terms
Find the sum of the first 25 terms of an A.P. whose nth term is given by an = 7 − 3n
If numbers n – 2, 4n – 1 and 5n + 2 are in A.P., find the value of n and its next two terms.
Find the sum of all multiples of 7 lying between 300 and 700.
Which term of the AP 3,8, 13,18,…. Will be 55 more than its 20th term?
Find the 6th term form the end of the AP 17, 14, 11, ……, (-40).
Is 184 a term of the AP 3, 7, 11, 15, ….?
Determine the nth term of the AP whose 7th term is -1 and 16th term is 17.
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?
Show that `(a-b)^2 , (a^2 + b^2 ) and ( a^2+ b^2) ` are in AP.
Find the sum of the following Aps:
9, 7, 5, 3 … to 14 terms
Write an A.P. whose first term is a and common difference is d in the following.
a = 6, d = –3
The sum of 5th and 9th terms of an A.P. is 30. If its 25th term is three times its 8th term, find the A.P.
If \[\frac{5 + 9 + 13 + . . . \text{ to n terms} }{7 + 9 + 11 + . . . \text{ to (n + 1) terms}} = \frac{17}{16},\] then n =
The common difference of the A.P. \[\frac{1}{2b}, \frac{1 - 6b}{2b}, \frac{1 - 12b}{2b}, . . .\] is
If an = 3 – 4n, show that a1, a2, a3,... form an AP. Also find S20.
The sum of the first five terms of an AP and the sum of the first seven terms of the same AP is 167. If the sum of the first ten terms of this AP is 235, find the sum of its first twenty terms.
If the first term of an AP is –5 and the common difference is 2, then the sum of the first 6 terms is ______.
Find the sum:
`4 - 1/n + 4 - 2/n + 4 - 3/n + ...` upto n terms
