If coefficients of three successive terms in the expansion of(x+1)nare...
To solve this problem, we need to use the concept of binomial expansion and the ratios of the coefficients. Let's break down the steps to find the value of 'n':
**Step 1: Understanding the Binomial Expansion**
The expansion of (x + 1)^n can be written as:
(x + 1)^n = C(n, 0) * x^n * 1^0 + C(n, 1) * x^(n-1) * 1^1 + C(n, 2) * x^(n-2) * 1^2 + ... + C(n, n) * x^0 * 1^n
Here, C(n, r) represents the binomial coefficient, which is given by the formula:
C(n, r) = n! / (r! * (n - r)!)
**Step 2: Understanding the Ratios**
According to the question, the coefficients of three successive terms are in the ratio 1 : 3 : 5. Let's assume the coefficients are a, b, and c, respectively.
So, we have:
a : b : c = 1 : 3 : 5
**Step 3: Finding the Ratios of Binomial Coefficients**
Using the formula for binomial coefficients, we can express the ratios of the coefficients in terms of the binomial coefficients:
a : b : c = C(n, 0) : C(n, 1) : C(n, 2)
We can simplify this further by using the formula for binomial coefficients:
a : b : c = 1 : n : (n * (n - 1)) / 2
**Step 4: Finding the Value of 'n'**
Now, we can equate the ratios of the coefficients to the given ratio:
1 : n : (n * (n - 1)) / 2 = 1 : 3 : 5
From this equation, we can find the value of 'n' by equating the corresponding terms:
n = 3
Therefore, the correct answer is option 'C' (7).