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 $a$ the next is $a+k$ the next is $a+2k$ and so on ... the ninth is $a+8k$.

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

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

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 $a$, and the common difference is $d = 1$.

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

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