Class 11

Math

Algebra

Sequences and Series

Insert $x$ arithmetic means between $x_{2}$ and $1$.

$a=x_{2}$ and $t_{x+2}=1$

$t_{n}=a+(n−1)d⇒t_{x+2}=a+(x+2−1)d⇒1=x_{2}+(x+1)d⇒1−x_{2}=(x+1)d⇒(x+1)(x−1)=(x+1)d⇒d=1−x$

So the arithmetic means are

So the arithmetic means are

$a+d,a+2d+a+3d...........a+(x+1)d⇒x_{2}−x+1,x_{2}−2x+2,.......,x$