Binomial Theorem for JEE: 30 flashcards
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1. State the Binomial Theorem for positive integer $n$.
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$$(a+b)^n = \sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^r$$
2. Formula for the binomial coefficient $\binom{n}{r}$?
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$\binom{n}{r} = \dfrac{n!}{r!(n-r)!}$
3. How many terms does $(a+b)^n$ have?
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$n+1$ terms
4. Write the general term $T_{r+1}$ of $(a+b)^n$.
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$T_{r+1} = \binom{n}{r}a^{n-r}b^r$
5. State Pascal's identity.
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$\binom{n}{r} = \binom{n-1}{r-1} + \binom{n-1}{r}$
6. Symmetry property of binomial coefficients?
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$\binom{n}{r} = \binom{n}{n-r}$
7. How many middle terms when $n$ is even?
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One middle term, $T_{n/2+1}$
8. How many middle terms when $n$ is odd?
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Two middle terms: $T_{(n+1)/2}$ and $T_{(n+3)/2}$
9. Sum of all binomial coefficients $\sum_{r=0}^{n}\binom{n}{r}$?
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$2^n$
10. Value of $\binom{n}{0} - \binom{n}{1} + \binom{n}{2} - \cdots$ for $n\ge1$?
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$0$
11. Sum of even-indexed coefficients equals what?
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$2^{n-1}$ (same as the sum of odd-indexed coefficients)
12. How do you find the term independent of $x$?
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Write $T_{r+1}$, set the exponent of $x$ to zero, solve for $r$.
13. Where does the greatest binomial coefficient occur for even $n$?
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At $r = n/2$
14. Where does the greatest binomial coefficient occur for odd $n$?
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At $r = (n-1)/2$ and $r = (n+1)/2$
15. How is the greatest term in an expansion found?
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Compare $\dfrac{T_{r+1}}{T_r}$ with 1 to see where terms stop increasing.
16. Base values $\binom{n}{0}$, $\binom{n}{1}$, $\binom{n}{n}$?
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$\binom{n}{0}=1$, $\binom{n}{1}=n$, $\binom{n}{n}=1$
17. Approximation of $(1+x)^n$ for small $x$?
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$(1+x)^n \approx 1 + nx$
18. For which index is the standard Binomial Theorem stated?
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Positive integral index $n$ (extensions for rational index are separate).
19. What does differentiating $(1+x)^n$ give?
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$\sum_{r} r\binom{n}{r}x^{r-1} = n(1+x)^{n-1}$
20. Term independent of $x$ in $\left(x^2+\frac{1}{x}\right)^9$?
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$\binom{9}{6} = 84$
21. Middle term of $(x+y)^6$?
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$T_4$
22. Middle terms of $(x+y)^7$?
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$T_4$ and $T_5$
23. Last term of $(a+b)^n$?
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$b^n$ (that is $T_{n+1}$)
24. Coefficient of $x^k$ in $(1+x)^n$?
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$\binom{n}{k}$
25. Which row of Pascal's triangle gives coefficients of $(a+b)^n$?
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Row $n$
26. Divisibility fact about $a^n - b^n$?
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$a^n - b^n$ is divisible by $a-b$ for all positive integers $n$.
27. How does the binomial distribution use the theorem?
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Its probabilities are terms of $(q+p)^n$: $\binom{n}{r}p^r q^{n-r}$.
28. What does integrating $(1+x)^n$ help compute?
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Sums involving $\dfrac{\binom{n}{r}}{r+1}$
29. Common mistake: is $\binom{n}{r}$ equal to $\frac{n}{r}$?
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No. $\binom{n}{r} = \dfrac{n(n-1)\cdots(n-r+1)}{r!}$ — only $r$ factors in the numerator.
30. How many terms in $(a+b)^{10}$?
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$11$ terms
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