Question

If the common difference in an arithmetic is twice the first term, show that Sn/Sm=n^2/m^2

Answer 1

Just to be clear, in an arithmetic progression, the sum until the nth term is given by...
\[\huge S_n = \frac{n(a_1+a_n)}{2}\]

Answer 2

ok

Answer 3

So, what have you tried so far?

Answer 4

i don't know how to start

Answer 5

Okay, it would help to have a formula for ...
\[\huge a_n = ?\]

Answer 6

In general, when given an arithmetic progression,
\[\huge a_n = a_1+(n-1)d\]where d is the common difference. We have an expression for the common difference, what is it?

Answer 7

its twice the first term

Answer 8

And the first term is \(\large a_1\)
Therefore...
\[\huge d = 2a_1\]
Substituting this...
\[\huge a_n =a_1+(n-1)(2a_1) \]
Please simplify this.

Answer 9

can u just give me an answer

Answer 10

@Mertsj

Answer 11

Nobody can just give you an answer because the question says to SHOW so why don't you go ahead and simplify the expression that terenz posted?

Answer 12

fine a1+(n-1)d

Answer 13

how does this help

Answer 14

He gave you the expression where he substituted for d per the problem instructions. That's the one you need to simplify

Answer 15

And what's with the hostile attitude toward the people who are trying to help you.

Answer 16

And I don't know how it will help...guess we'll find out as the problem progresses.

Answer 17

The asker has disappeared.

Answer 18

well thats annoying.. had the answer for him.. aaai

Answer 19

What is Sm?

Answer 20

i am back @Mertsj

Answer 21

@Mertsj Sm means the sum of all terms till the mth term.

Answer 22

can u help me lucky

Answer 23

I'll try to simplify the problem. Not guaranteed that i'll make it baby-ish.

Now As the main replier has correctly said -

Nth term = a + (n-1)2a
= a + 2an -2a => 2an - a => a(2n-1)
Mth term = a + (m-1)2a

=> 2am - a = a(2m-1)

Now sum till the Nth = N/2 ( a + a(2n-1))

Sum till the Mth = M/2 ( a + a(2m-1)

=> So Sm/Sn = N/2(a + 2an - a)/ M/2(a + 2am - a)

=> 2 is cancelled, a-a = 0

=> Sm/Sn = n(2an)/m(2am)

Simplify. :)

Answer 24

k thanks

Answer 25

i got Sn/Sm=n^2/m^2

Answer 26

That's what the directions said to show. Good job!!