If the coefficients of P th, $$($$ p $$+$$ $$1)th$$ and $$(p$$ $$+$$ $$2)th$$ terms in the expansion of $$(1$$ $$+$$ $$x)n$$ are in AP, then

Question

If the coefficients of  P th, $$($$ p $$+$$ $$1)th$$ and $$(p$$ $$+$$ $$2)th$$ terms in the expansion of $$(1$$ $$+$$ $$x)n$$ are in AP, then

Infinity Learn · Accepted Answer

coefficients of pth , (p$$ $$+$$ $$1)th$$ and $$(p$$ $$+$$ $$z)th$$ terms in the expansion $$(1$$ $$+$$ $$x)" are  nCp−$$1$$, nCp′   $$nCp+1$$, respectively.Since, these are in AP.∴ $$2nCp=nCp$$−$$1+nCp+1$$⇒$$2n$$!(n$$−$$p)!p!$$=n$$!(n$$−$$p+1)!(p$$−$$1)!$$+n$$!(n$$−p−$$1)!(p+1)! ⇒$$2(n$$−$$p)$$!p!$$=p(n$$−$$p+1)(n$$−$$p)$$!p!$$+n$$−$$p(n$$−$$p)$$!$$(p+1)p$$!        ⇒ $$21=p(n$$−$$p+1)+n$$−$$pp+1$$⇒ $$n2$$−$$n(4p+1)+4p2$$−$$2=0$$

If the coefficients of P th, ( p + 1)th and (p + 2)th terms in the expansion of (1 + x)ⁿ are in AP, then

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

free study material

If the coefficients of P th, ( p + 1)th and (p + 2)th terms in the expansion of (1 + x)n are in AP, then

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

free study material

If the coefficients of P th, ( p + 1)th and (p + 2)th terms in the expansion of (1 + x)ⁿ are in AP, then