Question

An arithmetic series contains n terms. Show that if t1 = a−b and tn = a+b then the value of Sn is independent of b.

[Arithmetic Sequences]

Answer 1

Sn is independent of b

Explanation:

t1=a-b

tn=a+b

we know that nth term of arithmetic series is an=a1+(n-1)d

so

a+b=a-b+(n-1)d

⇒2b=(n-1)d-----------equation 1

formula for sum of n terms of arithmetic series

Sn=
`(n)/(2)`(2a1+(n-1)d)

⇒Sn=
`(n)/(2)`(2(a-b)+2b) (since (n-1)d=2b from equation 1)

⇒Sn=
`(n)/(2)`(2a)

therefore we can see that Sn is independent of b