If the coefficients of the r th, $$(r$$ $$+$$ $$1)th$$ and $$(r+2)th$$ terms in the binomial expansion of $$(1$$ $$+$$ $$y)m$$ are in A.P., then m and r satisfy the equation

Question

If the coefficients of the r th, $$(r$$ $$+$$ $$1)th$$ and $$(r+2)th$$ terms in the binomial expansion of $$(1$$ $$+$$ $$y)m$$ are in A.P., then m and r satisfy the equation

Infinity Learn · Accepted Answer

Coefficient of the r th term is  mCr−$$1$$ According to the given condition  mCr−$$1+mCr+1=2$$ mCr⇒     mCr−$$1$$ $$mCr+$$ $$mCr+1$$ $$mCr=2$$⇒$$rm+1$$−$$r+m$$−$$rr+1=2$$⇒    $$r2+r+(m$$−$$r)(m+1$$−$$r)=2(m+1$$−$$r)(r+1)$$⇒    $$r2+r+m2+(1$$−$$2r)m$$−$$r(1$$−$$r)=$$    $$2m(r+1)+21$$−$$r2$$⇒    $$m2$$−$$m(4r+1)+4r2$$−$$2=0$$.

If the coefficients of the $r t h, (r + 1) t h$ and $(r + 2) t h$ terms in the binomial expansion of ${(1 + y)}^{m}$ are in A.P., then m and r satisfy the equation

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If the coefficients of the r th, (r + 1)th and (r+2)th terms in the binomial expansion of (1 + y)m are in A.P., then m and r satisfy the equation

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If the coefficients of the $r t h, (r + 1) t h$ and $(r + 2) t h$ terms in the binomial expansion of ${(1 + y)}^{m}$ are in A.P., then m and r satisfy the equation