Let the smallest value of k N , for which the coefficient of x^3 in (1+x)^3 + (1+x)^4 + (1+x)^5 + + (1+x)⁹⁹ + (1+kx)¹⁰⁰ , x 0 , is (43n + 101 4 ) (¹⁰⁰C₃ ) for some n N , be p . Then the value of p + n is:

Step-by-step solution

The coefficient of x^3 in the given expression is the sum of the coefficients of x^3 in each term. The coefficient of x^3 in _ r=3 ⁹⁹ (1+x)^r + (1+kx)¹⁰⁰ is: _ r=3 ⁹⁹ ^ r C₃ + k^3 ¹⁰⁰C₃ Using the identity _ r=k ^ n ^ r C_k = n+1 k+1 , we get: _ r=3 ⁹⁹ ^ r C₃ = ¹⁰⁰C₄ Thus, the total coefficient of x^3 is ¹⁰⁰C₄ + k^3 ¹⁰⁰C₃ . Given that this coefficient is equal to (43n + 101 4 ) ¹⁰⁰C₃ , we have: ¹⁰⁰C₄ + k^3 ¹⁰⁰C₃ = (43n + 101 4 ) ¹⁰⁰C₃ Dividing both sides by ¹⁰⁰C₃ : ¹⁰⁰C₄ ¹⁰⁰C₃ + k^3 = 43n + 101 4 Using ^ n C_r ^ n C