If for some positive integer nthe coefficients of three consecutive terms in the binomial expansion of $$(1+x)n+5are$$ in the ratio $$5$$: $$10$$; $$14$$,. then the largest $$coefficient-$$ in the expansion is

Question

If for some positive integer  nthe coefficients of three consecutive  terms in the binomial  expansion of $$(1+x)n+5are$$ in the ratio  $$5$$: $$10$$; $$14$$,. then the largest $$coefficient-$$ in the expansion is

Infinity Learn · Accepted Answer

Let three consecutive  terms be Tr,$$Tr+1$$ and $$Tr+2$$ Their coefficients are  $$n+5Cr$$−$$1$$,$$n+5Cr$$  and  $$n+5Cr+1$$ .It is given that  $$n+5Cr$$−$$1$$:$$n+5Cr$$:$$n+5Cr+1=5$$:$$10$$:$$14$$ $$n+5Cr$$ $$n+5Cr$$−$$1=105$$ and  $$n+5Cr+1$$ $$n+5Cr=1410$$⇒$$n+5$$−$$r+1r=2$$  and $$n+5$$−$$rr+1=75$$ ∵nrnr−$$1=n$$−$$r+1r$$⇒n−$$r+6=2r$$ and  $$5(n$$−$$r+5)=7r+7$$⇒n−$$3r+6=0$$ and $$5n$$−$$12r+18=0$$⇒$$n=6$$,$$r=4$$ Thus, the coefficients in the binomial expansion of $$(1+x)11$$ are  $$11C0$$,$$11C1$$,…,$$11C11$$.The greatest coefficient  is  $$11C5=11C6=462$$

If for some positive integer $n$ the coefficients of three consecutive terms in the binomial expansion of ${(1 + x)}^{n + 5}$ are in the ratio $5 : 10; 14, .$ then the largest coefficient- in the expansion is

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If for some positive integer nthe coefficients of three consecutive terms in the binomial expansion of (1+x)n+5are in the ratio 5: 10; 14,. then the largest coefficient- in the expansion is

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If for some positive integer $n$ the coefficients of three consecutive terms in the binomial expansion of ${(1 + x)}^{n + 5}$ are in the ratio $5 : 10; 14, .$ then the largest coefficient- in the expansion is