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

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

Shibashis Mallik , 10 Years ago

Grade 12

4 Answers

SHAIK AASIF AHAMED

Last Activity: 10 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+..............+a_2m-1²-a_2m²
=(a1+a2)(a1-a2)+.............(a_2m-1+a_2m)(a_2m-1-a_2m)
=-d(a1+a2+.........+a2m)
=-d(2n/2)(a1+a2m)
=-dm(a₁²-a_2m²)
=dm(a₁²-a_2m²)/(a_2m-a₁)
=(m/(2m-1))[a₁²-a_2m²] since [a_2m=a1+(2m-1)d]

Approved

Shibashis Mallik

Last Activity: 10 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

Last Activity: 10 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+..............+a_2m-1²-a_2m²
=(a1+a2)(a1-a2)+.............+(a_2m-1+a_2m)(a_2m-1-a_2m)
= -d(a₁+a₂+.........+a_2m) since a1-a2=a2-a3=.....=a_2m-1-a_2m=-d
=-d(2m/2)(a₁+a_2m)
[since sum to n terms =n/2(a+l) here there are 2m terms where l=last term]
=-dm(a₁²-a_2m²)/(a₁-a_2m)
=dm(a₁²-a_2m²)/(a_2m-a₁)
=(m/(2m-1))[a₁²-a_2m²] since [a_2m=a1+(2m-1)d which is nth term formula]
I hope you will understand this now