Help with further pure 1- summing series?

MEPS1996

Hello, i wandered if anybody could help me with this problem in summing series using the difference method,

show that ra^r-(r-1)a^(r-1)=(a-1)ra^(r-1)+a^(r-1) and use this result to find sum(ra^(r-1)) with limits 1 to n.

I know the questions looks nasty but i would really appreciate any help with this.
Thanks

Original post by MEPS1996

Hello, i wandered if anybody could help me with this problem in summing series using the difference method,

show that ra^r-(r-1)a^(r-1)=(a-1)ra^(r-1)+a^(r-1) and use this result to find sum(ra^(r-1)) with limits 1 to n.

I know the questions looks nasty but i would really appreciate any help with this.
Thanks

I was about to help but that is really difficult to read.

Would you mind using LaTex?

It's ok. I've figured what you mean out now!

Note that

a^r = a (a^{r-1})

and then factorise

a^{r-1}

from the left hand side.

Reply 3

MEPS1996

OP

4

Original post by Mr M

It's ok. I've figured what you mean out now!

Note that

a^r = a (a^{r-1})

and then factorise

a^{r-1}

from the left hand side.

Sorry for late reply but do you know how to do the second part of this question where you have to find the sum of ra^(r-1) from 1 to n?

Reply 4

joostan

13

Original post by MEPS1996

Sorry for late reply but do you know how to do the second part of this question where you have to find the sum of ra^(r-1) from 1 to n?

Since MrM is offline:
I'm not 100% what your series is, but I imagine since you broke it into two series that the method of differences is what you're after. :smile:

Reply 5

Ateo

9

If it helps anyone:

\displaystyle \sum^{n}_{r=1} \left[ ra^r - (r-1)a^{r-1} \right] = \sum^n_{r=1} \left[ (a-1)ra^{r-1} + a^{r-1} \right]

Reply 6

Ateo

9

In order to find the required series, you want to use telescopic series to simplify the LHS of my post above. You can then use the sum of a geometric series on the right-most term - you then take it over to LHS. Now you have an equation with a sum on LHS and the (a-1) x the required series on the right. So you simply divide by (a-1) to get your final answer. Can you progress through the question with that?

(Type your working in Latex)

Help with further pure 1- summing series?

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