|Coefficient of a particular power of x|
Eample: Find the co-efficient of x152)10.
- Let the particular power occur in the (r+1)th term.
- Write the (r+1)th term of the given binomial.
- Equate the power of x in the (r+1)th term and the given power.
- Evaluate r and substitute in in step 2.
Terms Independent of x
- Let x15 occur in the (r+1)th term.
- Tr+1 = 10Crx10-r .(-x2)r.
- = (-1)r.10Crx10+r
- x10+r = x15
- r = 5
- Let (r+1)th term be the term independent of x.
- Put the power of x in this term equal to zero and evaluate it.
Example: Find the term independent of x in the expansion of (3/2x2 - 1/3x)9.
- Let (r+1)th term be independent of x.
- Tr+1 = 9Cr.(3/2x2)9-r.(-1/3x)r
- Putting 18-3r = 0, we get r = 6.
- The required term is (-1)6.9C6. 3-3/23 = 9C3.(1/33.23 = 7/18
Example: Find the greatest coefficient in the expansion of (1+x)10
Answer : 252
- Let Tr+1 have the greatest coefficient.
- Then coefficient of Tr+1 >= the coefficient of T.
- 10Cr >= 10Cr-1
- (10-r+1)/r >= 1 or 11 >= 2r, or r <= 51/2
- Hence upto r = 5 the coefficients increase, the greatest of them being r = 5.
- The greatest coefficient = 10C5 = 252
Directions: Answer the following