Class 10 Math All topics Arithmetic Progressions

If $S_{r}$ denotes the sum of $r$ terms of an AP and $a_{2}S_{a} =b_{2}S_{b} =c$ then $S_{c}$ is

(a)

$c_{3}$

(b)

$c/ab$

(c)

$abc$

(d)

$a+b+c$

Correct answer: (a)

Solution: $S_{r}$ denotes sum of $r$ terms.

$a_{2}S_{a} =b_{2}S_{b} a_{2}2a (2a_{1}+(a−1)d) =b_{2}2b (2a_{1}+(b−1)d) a2a_{1} +a(a−1)d =b2a_{1} +b(b−1)d or,a2a_{1} +d−ad =b2a_{1} +d−bd or,2a_{1}(a1 −b1 )=d(a1 −b1 )So,2a (a_{2}2a_{1}+(a−1)d )=cor,2a_{1}2a(1+a−a) =caa_{1}a =cc=a_{1}S_{c}=2c (2a_{1}+(c−1)d)=2c (2a_{1}+(c−1)2a_{1})=2c (2c+(c−1)2c)=2c ×2c(1+c−1)=c_{3}$

Answer $(A)$

Similar topics

introduction to trigonometry

functions

some applications of trigonometry

quadratic equations

surface areas and volumes

introduction to trigonometry

functions

some applications of trigonometry

quadratic equations

surface areas and volumes

Related Questions

Related Questions