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Let a1, a2, a3, ….. be an A.P. Prove that

Let a1, a2, a3, ….. be an A.P. Prove that

Question Image
Grade:12

4 Answers

SHAIK AASIF AHAMED
askIITians Faculty 74 Points
6 years ago
Hello student,
Please find the answer to your question below
Let d be the common difference in AP
So d=a2-a1=a3-a2=a4-a3=........=a2m-a2m-1
Now a12-a22+..............+a2m-12-a2m2
=(a1+a2)(a1-a2)+.............(a2m-1+a2m)(a2m-1-a2m)
=-d(a1+a2+.........+a2m)
=-d(2n/2)(a1+a2m)
=-dm(a12-a2m2)
=dm(a12-a2m2)/(a2m-a1)
=(m/(2m-1))[a12-a2m2] since [a2m=a1+(2m-1)d]
Shibashis Mallik
21 Points
6 years ago
I didn’t understood this step )1-a2m)/(a22m-a21)=dm(a22m-a21-dm(a
 
plzz...explain
And also in this step
)=-d(a1+a2+.........+a2m)2m-a2m-1)(a2m+a2m-1=(a1+a2)(a1-a2)+.............(a
I think it should be -d(a1 + 2a2 + 2a3 + …...+ a2m)
SHAIK AASIF AHAMED
askIITians Faculty 74 Points
6 years ago
Hello student,
Please find the answer to your question below
Iam explaining every step again here
Let d be the common difference in AP
So d=a2-a1=a3-a2=a4-a3=........=a2m-a2m-1
Now a12-a22+..............+a2m-12-a2m2
=(a1+a2)(a1-a2)+.............+(a2m-1+a2m)(a2m-1-a2m)
= -d(a1+a2+.........+a2m) since a1-a2=a2-a3=.....=a2m-1-a2m=-d
=-d(2m/2)(a1+a2m)
[since sum to n terms =n/2(a+l) here there are 2m terms where l=last term]
=-dm(a12-a2m2)/(a1-a2m)
=dm(a12-a2m2)/(a2m-a1)
=(m/(2m-1))[a12-a2m2] since [a2m=a1+(2m-1)d which is nth term formula]
I hope you will understand this now
Shibashis Mallik
21 Points
6 years ago
Thankx a lot for explaining me...

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