Evaluate $C_0+4C_1+7C_2+...+ (3n+1)C_n - Math Stack Exchange

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Learn more about Teams Evaluate $C_0+4C_1+7C_2+...+ (3n+1)C_n$ Ask Question Asked 11 years, 10 months ago Modified 11 years, 10 months ago Viewed 628 times 0 $\begingroup$

Evaluate $C_0+4C_1+7C_2+...+ (3n+1)C_n$ where $C_n$ are binomial coefficients as we know.

I get the answer as $3n2^{n-1}+2^n$ .am I correct? Please somebody ensure me please. thanks for your help.

• combinatorics

Share Cite Follow asked Jan 20, 2013 at 11:31 ramarama 1122 bronze badges $\endgroup$ 1

• $\begingroup$ Maybe you may know, but the binomial coefficients that I know of have two indices, not one, and are written $\binom nk$. In other words your question does not seem to make sense. $\endgroup$ – Marc van Leeuwen Commented Jan 20, 2013 at 11:44

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Sorted by: Reset to default Highest score (default) Date modified (newest first) Date created (oldest first) 1 $\begingroup$

$$T_{r+1}=(3r+1)\binom nr=3r\frac{n!}{(n-r)! r!}+\binom nr \text{where}\space 0\le r\le n$$

Now, $$r\frac{n!}{(n-r)! r!}=r\frac{n!}{\{n-1-(r-1)\}!) r(r-1)!} \text{if} \space r\ge1$$

$$r\frac{n!}{(n-r)! r!}=n\frac{(n-1)!}{\{n-1-(r-1)\}!) (r-1)!} =n\binom{n-1}{r-1}$$

$$\sum_{0\le r\le n}(3r+1)\binom nr$$ $$=3n\sum_{0\le r\le n}\binom{n-1}{r-1}+\sum_{0\le r\le n}\binom nr$$ $$=3n\sum_{1\le r\le n}\binom{n-1}{r-1}+\sum_{0\le r\le n}\binom nr$$ $$=3n(1+1)^{n-1}+(1+1)^n=3n2^{n-1}+2^n=2^{n-1}(3n+2)$$

Share Cite Follow answered Jan 20, 2013 at 11:41 lab bhattacharjeelab bhattacharjee 277k2020 gold badges207207 silver badges328328 bronze badges $\endgroup$ 2

• $\begingroup$ that's mean I am right.thanks for your help. $\endgroup$ – rama Commented Jan 20, 2013 at 11:45
• $\begingroup$ @rama, welcome. Btw, how have you calculated? $\endgroup$ – lab bhattacharjee Commented Jan 20, 2013 at 11:47

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