the sum of the four terms of an Arithmetic progression is 40 And product of second term and third term is 96. find Arithmetic progression
Answers
Step-by-step explanation:
Given:-
The sum of the four terms of an Arithmetic Progressionrogression is 40 and product of second term and third term is 96.
To find:-
Find Arithmetic progression ?
Solution:-
Let the first term of an AP = a
Common difference = d
We know that
The sum of first n terms in an AP
Sn = (n/2)[2a+(n-1)d]
Given that
The sum of the four terms of an Arithmetic Progressionrogression is 40
=> S4 = (4/2)[2a+(4-1)d] = 40
=> S4 = (2)[2a+3d] = 40
=> 2a+3d = 40/2
=> 2a +3d = 20
=> 2 a = 20-3d
=> a = (20-3d)/2 --------(1)
We know that
The general term of an AP = an = a+(n-1)d
Now,
Second term = a+d
Third term = a+2d
Given that
Product of the second and third terms = 96
=> (a+d)(a+2d) = 96
=>[(20-3d)/2 +d] [(20-3d)/2 +2d ] = 96
=> [(20-3d+2d)/2][(20-3d+4d)/2] = 96
=> [(20-d)/2][(20+d)/2] = 96
=> (20-d)(20+d)/4 = 9
=> (20-d)(20+d) = 96×4
=>(20-d)(20+d) = 384
=> 20^2 -d^2 = 384
(Since (a+b)(a-b)=a^2-b^2)
=> 400-d^2 = 3840
=> -d^2 = 384-400
=> -d^2 = -16
=> d^2 = 16
=> d = ±√16
=>d=±4
=> d = 4 or -4
If d = 4 then
a = [20-3(4)]/2
=> a = (20-12)/2
=> a = 8/2
=> a = 4
if d = -4 then
a =[20-3(-4)]/2
=> a = (20+12)/2
=> a = 32/2
=> a = 16
The general form of the AP : a ,a+d,a+2d,...
If a = 4 and d = 4 then AP
4, 4+4, 4+2(4),4+3(4)...
=> 4 ,8,12,16,...
If a = 16 and d = -4 then AP
16, 16-4 , 16+2(-4),...
=> 16, 12 , 8,..
Answer:-
The required Arithmetic Progression is 4,8,12,16,... or 16,12,8,4,....
Used formulae:-
1.The general form an AP is a,a+d,a+2d,...
2.The general term of an AP
= an = a+(n-1)d
3.The sum of first n terms in an AP
Sn = (n/2)[2a+(n-1)d]
4.(a+b)(a-b)=a^2-b^2