1. ## arithmetic sequence

hello
'find the arithmetic progression of nine continuous numbers where the sum total is 135'
kindly get me detailed solution
sincerely
anil

hello
'find the arithmetic progression of nine continuous numbers where the sum total is 135'
kindly get me detailed solution
sincerely
anil
The first number is $\displaystyle a$ the next is $\displaystyle a+k$ the next is $\displaystyle a+2k$ and so on ... the ninth is $\displaystyle a+8k$.

In sum we have,
$\displaystyle \underbrace{(a+a+...+a)}_9 + k(1+2+...+8) = 9a+36k$.

So we want,
$\displaystyle 9a+36k=135$
Divide by 9,
$\displaystyle a+4k=15$.
So the possible such sequences are:
$\displaystyle a=11,k=1$
$\displaystyle a=7,k=2$
$\displaystyle a=3,k=3$

3. Hello, Anil!

Find the arithmetic progression of nine continuous numbers where the sum total is 135
Did you mean "consecutive" numbers?
. . Then the first term is $\displaystyle a$, and the common difference is $\displaystyle d = 1$.

The sum of the first n terms of an A.P. is: .$\displaystyle S_n\:=\:\frac{n}{2}\left[2a + (n-1)d\right]$
. . Then we have: .$\displaystyle S_9 \:=\:\frac{9}{2}\left[2a + 8(1)\right] \:=\:135\quad\Rightarrow\quad a \:=\:11$

Therefore, the progression is: .$\displaystyle 11,\,12,\,13,\,14,\,15,\,16,\,17,\,18,\,19$