Let the first term be a ,
The common difference be d .
Then the sum of first three terms = a+a+d+a+2d = 3a+3d .
Given 3(a+d) = 33
=> a+d = 11 .
=> d =11-a
Therefore ,Second term of A.P = 11 .
The product of first and third terms = (a)(a+2d) = a(a+2(11-a)
= a(a+22-2a)
= a(22-a)
= 22a-a²
ATQ --->
Given 22a-a²-29= a+d
=> 22a-a²-29=11
=> 22a-a² -40 =0
=> a²-22a+40=0
=> a²-20a-2a+40=0
=> a(a-20)-2(a-20) =0
=> a= 2 or 20.
Finding common difference for a = 2
11-2=9
Finding Common difference for a =20
11-20=-9 .
Now The possible A .P 's are
1) 2,11,20,29,38,47,56,65,74.....
2) 20,11,2,-7,-16,-25,-34,-43,-52,-61,-70 .......
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Answers & Comments
a-d+a+a+d=33
3a=33
a=11
(a-d)(a+d)=a+29
a²-d²=a+29
121-d²=11+29
121-d²=40
d²=81
d=+/-9
When d=+9
The terms are 2, 11 and 20
When d=-9
The terms are 20,11 and 2
Let the first term be a ,
The common difference be d .
Then the sum of first three terms = a+a+d+a+2d = 3a+3d .
Given 3(a+d) = 33
=> a+d = 11 .
=> d =11-a
Therefore ,Second term of A.P = 11 .
The product of first and third terms = (a)(a+2d) = a(a+2(11-a)
= a(a+22-2a)
= a(22-a)
= 22a-a²
ATQ --->
Given 22a-a²-29= a+d
=> 22a-a²-29=11
=> 22a-a² -40 =0
=> a²-22a+40=0
=> a²-20a-2a+40=0
=> a(a-20)-2(a-20) =0
=> a= 2 or 20.
Finding common difference for a = 2
11-2=9
Finding Common difference for a =20
11-20=-9 .
Now The possible A .P 's are
1) 2,11,20,29,38,47,56,65,74.....
2) 20,11,2,-7,-16,-25,-34,-43,-52,-61,-70 .......