Binomial Theorem for Positive Integral Indices

Tips :

• If ‘n’ is a positive integer, then (x + a)n = nCo . xn + nC1. xn-1 . a + nC2. xn-2 . a2 + ....... + nCn . an
• nCo = nCn ; nCr = nCn-r
• If ‘n’ is odd, there are two greatest binomial coefficients in the expansion which are \tt ^n C _\frac{n-1}{2}\ and\ ^n C _\frac{n+1}{2}
• If ‘n’ is even there is only one greatest binomial coefficient in the expansion (x + a)n which is \tt ^n C _\frac{n}{2}
• If n > 2, n ∈ N then (2n −1)n + (2n)n < (2n+1)n
• If the coefficients of Xr-1 , Xr , Xr+1 in (1 + x)n are in A.P then (n −2r)2 = n + 2

Tricks :

• Coefficient of Xp Yq Zr in (ax + by + cz)n is \tt \frac{n! a^{p}b^{q}c^{r}}{p! \times q! \times r!} where p+q+r = n and n, p, q, r ∈ N.
• (x + a)n – (x - a)n = 2 [nC1 xn−1 a + nC3 xn−3 a3 + nC5 xn−5 a5 + ...........]

Part5: View the Topic in this video From 00:10 To 07:30

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Binomial Theorem for Positive Integer:
If n is any positive integer, then (x + a)n = nC0 xn + nC1 xn−1 + a+ nC2 xn−2 a2 + ... + nCn an.
i.e., (x + a)^n \sum_{r=0}^n \ ^nC_r\ x^{n-r}a^{r}

2. (x − a)n = nC0nC1 xn−1 a + nC2 xn−2 a2nC3 xn−3 a3 + ... + (−1)n. nCn an
i.e., (x-a)^n \sum_{r=0}^n \ \left(-1\right) \ ^nC_r \cdot x^{n-r}\cdot a^{r}

3. (1 + x)n = nC0 + nC1 x  + nC2 x2  + ... +  nCn xn
i.e., (1+x)^n=\sum_{r=0}^n \ ^nC_r \cdot x^{r}

4. (x − a)n = nC0nC1 x  + nC2 x2  − nC3 x3 + ... + \left(-1\right)^{r^{n}}C_{r}x^{r} + ...+ (-1)n nCn xn
i.e.,(1-x)^n \sum_{r=0}^n \left(-1\right)^r \ {^{n}C_{r}}\cdot x^{r}

5. (x + a)n + (x − a)n = 2 (nC0xn a0 + nC2 xn−2 a2 + ...)
(x + a)n − (x − a)n = 2 (nC1xn−1 a + nC3 xn−3 a2 + ...)