Answer :

Given,


Sp = 1 + rp + r2p + … to ∞


We observe that the above progression possess a common ratio. So it is a geometric progression.


Common ratio = rp and first term (a) = 1


Sum of infinite GP = ,where a is the first term and k is the common ratio.


Note: We can only use the above formula if |k|<1


As, |r|<1 |rp|<1 if (p>1)


we can use the formula for the sum of infinite GP.


Sp = ….equation 1


As, sp = 1 – rp + r2p - … to ∞


We observe that the above progression possess a common ratio. So it is a geometric progression.


Common ratio = -rp and first term (a) = 1


Sum of infinite GP = ,where a is the first term and k is the common ratio.


Note: We can only use the above formula if |k|<1


As, |r|<1 |rp|<1 if (p>1)


we can use the formula for the sum of infinite GP.


sp = ….equation 2


As we have to prove - sp + Sp = 2 s2p


From equation 1 and 2, we get-


Sp + sp =


Sp + sp = {using (a+b)(a-b)=a2-b2}


Sp + sp =


As Sp =


following the same analogy, we have-



Sp + sp =


Hence,


Sp + sp = 2S2p


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