Question

if the sum of first n terms of ap is n(n+10) then find first term and common difference

Answer 1

sn=n/2(2a+(n-1)d)

i.e. s1=11

s1=a1

s2=2(2+10)

=2(12)

=24

s2=2/2(2*11+(2-1)d)

24=22+d

24-22=d

d=2

comman difference =2

a2=11+(2-1)2

a2=11+2

=13

Answer 2

This needs no more equations and methods to find them.

I'll prefer you a very short method which is the most suitable and the most helpful method for such questions. Who doesn't like to solve a math problem very quickly?!

I recommend you this method which is shown below. Maybe you can be appreciated by this method if this method is not learnt!!! ;-)

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TO REMEMBER...

                     

The sum of first $n$ terms of an AP is in the form of,

$(\frac{d}{2})n^2+(a-\frac{d}{2})n$

where $a$ = first term and $d$ = common difference,

which includes that,

the coefficient of $n^2$ is half of the common difference,

and the coefficient of $n$ is half of the common difference subtracted from the first term.

So,

Common difference $d$ = Twice the coefficient of $n^2$

&

First term $a$ = sum of coefficients of both $n^2$ and $n$

$[\because\ \frac{d}{2}+(a-\frac{d}{2})=\frac{d}{2}+a-\frac{d}{2}=a]$

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Now let's start.

Here, sum is n(n + 10).

$S_n = n(n + 10) \\ \\ S_n=n^2+10n$

So,

$\frac{d}{2}=1 \\ \\ a-\frac{d}{2}=10$

From coefficient of $n^2$,

$\frac{d}{2}=1 \\ \\ d=1 \times 2 \\ \\ d=\bold{2}$

By finding sum of coefficients of both,

$a=1+10 \\ \\ a=\bold{11}$

So that's all!

Only do this method for such questions.

And there's no need to find the first term and common difference, like by giving 1 as value for n to get the first term, then 2 as value for n to get the sum of first and second terms, then taking their difference to get second term, and then taking difference of first term and second term to get the common difference. This may take a few while.

           

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Hope this may be helpful.

If this may be helpful, don't forget to mark my answer as the brainliest.

If you've any doubt, ask me in the 'comments' section and I'll be able to clear it.

And sorry for the huge answer if it feels despair for you.

     

Thank you. Have a nice day. :-))

 

     

#adithyasajeevan