Home
Class 12
MATHS
Find the value of ^4nC0+^(4n)C4+^(4n)C8...

Find the value of ` ^4nC_0+^(4n)C_4+^(4n)C_8++""^(4n)C_(4n)` .

Text Solution

Verified by Experts

The correct Answer is:
`2^(4n-2)+(-1)^(n)2^(2n-1)`
We have,
`.^(4n)C_(0) + .^(4n)C_(2)x^(2)+.^(4n)C_(4)x^(4)+"…."+.^(4n)C_(4n)x^(4n)`
`= 1/2[(1+x)^(4n) +(1-x)^(4n)]`
Putting `x = 1` and `x= i`, we get
`.^(4n)C_(0) + .^(4n)C_(2) + .^(4n)C_(4) + "……" + .^(4n)C_(4n)=1/2[2^(4n)]`
and `.^(4n)C_(0) - .^(4n)C_(2) + .^(4n)C_(4) - "....." + .^(4n)C_(4n)`
`= 1/2[(1+i)^(4n)+(1-i)^(4n)]`
Thus, `2[.^(4n)C_(0) + .^(4n)C_(4)+"......"+.^(4n)C_(4n)] = 2^(4n-1) + 1/2[sqrt(2)(cos'(pi)/(4)- isin'(pi)/(4))]^(4n)`
`= 2^(2n) (cosnpi + isin npi)+2^(2n) (cos npi - i sin npi)`
`= 2^(2n+1) cos n pi=2^(2n+1)(-1)^(n)`
`:. 2[.^(4n)C_(0) + .^(4n)C_(4) + "....." + .^(4n)C_(4n)] = 2^(4n-1) + 1/2 2^(2n+1)(-1)^(n)`
`rArr .^(4n)C_(0) + .^(4n)C_(4) + "....." + .^(4n)C_(4n) = 2^(4n-2) + (-1)^(n)2^(2n-1)`
Promotional Banner

Topper's Solved these Questions

  • BINOMIAL THEOREM

    CENGAGE ENGLISH|Exercise Concept Application Exercise 8.5|8 Videos

  • BINOMIAL THEOREM

    CENGAGE ENGLISH|Exercise Concept Application Exercise 8.6|10 Videos

  • BINOMIAL THEOREM

    CENGAGE ENGLISH|Exercise Concept Application Exercise 8.3|7 Videos

  • AREA

    CENGAGE ENGLISH|Exercise Comprehension Type|2 Videos

  • CIRCLE

    CENGAGE ENGLISH|Exercise MATRIX MATCH TYPE|7 Videos

Similar Questions

Explore conceptually related problems

Find the value of ^(4n)C_0+^(4n)C_4+^(4n)C_8+….+^(4n)C_(4n)

Find n if ""^(n)C_(4)=""^(n)C_(6)

The value of |1 1 1^n C_1^(n+2)C_1^(n+4)C_1^n C_2^(n+2)C_2^(n+4)C_2| is

Find the value of n such that "(i) "^(n)P_(5) =42 xx ""^(n)P_(3), n gt 4 " (ii) "(""^(n)P_(4))/(" "^(n-1)P_(4))=(5)/(3), n gt 4.

If ""^(n)C_4= ""^(n)C_6, find n.

Find the following sums : (i) .^(n)C_(0)-.^(n)C_(2)+.^(n)C_(4)-.^(n)C_(6)+"....." (ii) .^(n)C_(1)-.^(n)C_(3)+.^(n)C_(5)-.^(n)C_(7)+"...." (iii) .^(n)C_(0)+.^(n)C_(4)+.^(n)C_(8)+.^(n)C_(12)+"....." (iv) .^(n)C_(2) + .^(n)C_(6) + .^(n)C_(10)+.^(n)C_(14)+"......" (v) .^(n)C_(1) + .^(n)C_(5)+.^(n)C_(9)+.^(n)C_(13)+"...." (vi) .^(n)C_(3) + .^(n)C_(7) + .^(n)C_(11) + .^(n)C_(15) + "....."

Find the sum .^n C_0+^n C_4+^n C_8 + . . .

Find the sum sum_(j=0)^n( ^(4n+1)C_j+^(4n+1)C_(2n-j)) .

If .^nC_30=^nC_4 , find n

If ""^(n)C_(3) + ""^(n)C_4 gt ""^(n+1) C_3 , then.