find the coefficient of x^29 in the expansion (1+2x)^12 .(1+x)^18
(2x+1)^12
= (2x)^12 + 12(2x)^11 + ...
= 2^12 x^12 + 12*2^11 x^11 + ...
(x+1)^18 = x^18 + 18x^17 + 9*17 x^16 + ...
= 2^12 x^30 + (18*2^12 + 12)x^29 + ...
so, 18*2^12 + 12 = 73740
(2x+1)^12 = 2^12 x^12 + 12*2^11 x^11 + ...
(x+1)^18 = x^18 + 18x^17 + ...
multiply the two expansions to get
2^12 x^30 + (18*2^12 + 12*2^11) x^29 + ...
so, we have 18*2^12 + 12*2^11 = 98304
To find the coefficient of x^29 in the expansion of (1+2x)^12 . (1+x)^18, we can use the binomial theorem.
The binomial theorem states that for any two numbers a and b, and any positive integer n, the expansion of (a + b)^n can be written as the sum of the terms of the form C(n, k) * a^(n-k) * b^k, where C(n, k) represents the binomial coefficient.
In this case, we have (1+2x)^12 and (1+x)^18, which can be expanded separately using the binomial theorem.
Let's start with (1+2x)^12. Apply the binomial theorem:
(1+2x)^12 = C(12, 0) * 1^12 * (2x)^0 + C(12, 1) * 1^11 * (2x)^1 + C(12, 2) * 1^10 * (2x)^2 + ... + C(12, 11) * 1^1 * (2x)^11 + C(12, 12) * 1^0 * (2x)^12
Simplifying this expression gives:
(1+2x)^12 = 1 + 24x + 264x^2 + ... + 435456x^11 + 4096x^12
Now let's expand (1+x)^18:
(1+x)^18 = C(18, 0) * 1^18 * x^0 + C(18, 1) * 1^17 * x^1 + C(18, 2) * 1^16 * x^2 + ... + C(18, 17) * 1^1 * x^17 + C(18, 18) * 1^0 * x^18
Simplifying this expression gives:
(1+x)^18 = 1 + 18x + 153x^2 + ... + 38760x^17 + 262144x^18
Now, we need to find the term with x^29 in the product of these two expressions. To obtain x^29, we need to have one x^12 term multiplied by one x^17 term.
There are multiple combinations that can give us x^29. We can have (24x^12) * (38760x^17) or (264x^2) * (262144x^18), among others.
Calculating the coefficient for each combination, we get:
Coefficient of x^29 = Coefficient of (24x^12) * (38760x^17) + Coefficient of (264x^2) * (262144x^18) + ...
To determine the coefficient, we need to multiply the corresponding coefficients:
Coefficient of x^29 = 24 * 38760 + 264 * 262144 + ...
We find all the terms that contain x^29 in the expansion and add up their coefficients.
To summarize:
1. Expand (1+2x)^12 using the binomial theorem.
2. Expand (1+x)^18 using the binomial theorem.
3. Identify the terms that contain x^29 in the expansion.
4. Multiply their coefficients.
5. Add up all the resulting coefficients to find the final coefficient of x^29.