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If the coefficients of the three successive terms in the binomial expansion of (1 + x)n are in the ratio 1 : 7 : 42, then the first of these terms in the expansion is :
- a)6th
- b)7th
- c)8th
- d)9th
Correct answer is option 'B'. Can you explain this answer?
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If the coefficients of the three successive terms in the binomial expa...
Solution:
We know the binomial expansion of (1 + x)n, which is given by:
(1 + x)n = nC0 + nC1x + nC2x2 + … + nCrxr + … + nCnxn
Here, the coefficients of three successive terms are in the ratio 1 : 7 : 42. Let us assume that the first term is a, the second term is 7a, and the third term is 42a.
Therefore, the coefficients of the three successive terms are nC0, nC1, and nC2 respectively.
We know that nCr = nC(n-r)
Therefore, nC1 = nC(n-1)
Also, nC2 = nC(n-2)
Now, we can write:
7a = 7nC1x
42a = 42nC2x2
Dividing equation (2) by equation (1), we get:
6 = 6n-1
n = 8
Therefore, the first term in the expansion is nC0, which is the coefficient of x0 and is equal to 1.
Hence, the first term in the expansion is the 7th term in the expansion.
Therefore, the correct answer is option (B).
We know the binomial expansion of (1 + x)n, which is given by:
(1 + x)n = nC0 + nC1x + nC2x2 + … + nCrxr + … + nCnxn
Here, the coefficients of three successive terms are in the ratio 1 : 7 : 42. Let us assume that the first term is a, the second term is 7a, and the third term is 42a.
Therefore, the coefficients of the three successive terms are nC0, nC1, and nC2 respectively.
We know that nCr = nC(n-r)
Therefore, nC1 = nC(n-1)
Also, nC2 = nC(n-2)
Now, we can write:
7a = 7nC1x
42a = 42nC2x2
Dividing equation (2) by equation (1), we get:
6 = 6n-1
n = 8
Therefore, the first term in the expansion is nC0, which is the coefficient of x0 and is equal to 1.
Hence, the first term in the expansion is the 7th term in the expansion.
Therefore, the correct answer is option (B).
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If the coefficients of the three successive terms in the binomial expansion of (1 + x)n are in the ratio 1 : 7 : 42, then the first of these terms in the expansion is :a)6thb)7thc)8thd)9thCorrect answer is option 'B'. Can you explain this answer?
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