The Total solution for NCERT class 6-12
If Sp┬аdenotes the sum of the series 1 + rp┬а+r2p┬а+ тАж to тИЮ and sp┬аthe sum of the series 1 тАУ rp┬а+r2p┬атАУ тАж to тИЮ, prove that sp┬а+ Sp┬а=2 S2p.
Given:
Sp┬а= 1+ rp┬а+ r2p┬а+ тАж тИЮ
By using the formula,
SтИЮ┬а=a/(1 тАУ r)
Where, a = 1, r = rp
So,
Sp┬а= 1/ (1 тАУ rp)
Similarly, sp┬а=1 тАУ rp┬а+ r2p┬атАУ тАж тИЮ
Where, a = 1, r = -rp
Sp┬а= 1/ (1 тАУ (-rp))
= 1 / (1 + rp)
Now, Sp┬а+sp┬а= [1 / (1 тАУ rp)] + [1 / (1 + rp)]
2S2p┬а=[(1 тАУ rp) + (1 + rp)] / (1 тАУ r2p)
= 2 /(1 тАУ r2p)
тИ┤ 2S2p┬а=Sp┬а+ Sp