To find the greatest term in the expansion of (2 + 3x)^9 when x = 3/2, we can use the Binomial Theorem. The Binomial Theorem provides a formula for expanding a binomial raised to a positive integer power.

The Binomial Theorem states that for any positive integer n, the expansion of (a + b)^n can be written as:

(a + b)^n = C(n, 0)a^n b^0 + C(n, 1)a^(n-1) b^1 + C(n, 2)a^(n-2) b^2 + ... + C(n, n-1)a^1 b^(n-1) + C(n, n)a^0 b^n

Where C(n, k) represents the binomial coefficient, given by:

C(n, k) = n! / (k!(n-k)!)

Now, let's apply the Binomial Theorem to the given expression and find the term with the greatest coefficient.



In the expansion of (2 + 3x)^9, the term with the greatest coefficient will have the largest binomial coefficient. This occurs when the exponent of x is at its highest value.

We can determine the exponent of x in each term by observing that the exponent of x decreases by 1 with each term, starting from 9 and ending at 0.

So, the term with the greatest coefficient will have the exponent of x equal to 9.



To calculate the term with the exponent of x equal to 9, we substitute the value of x = 3/2 into the expression (2 + 3x)^9 and simplify.

(2 + 3x)^9 = (2 + 3(3/2))^9
= (2 + 9/2)^9
= (4/2 + 9/2)^9
= (13/2)^9



To simplify the expression (13/2)^9, we raise the numerator and denominator of the fraction to the power of 9.

(13/2)^9 = 13^9 / 2^9

Now, let's calculate the values of 13^9 and 2^9 separately.

13^9 = 254,186,856,935,728
2^9 = 512

Therefore, (13/2)^9 = 254,186,856,935,728 / 512
= 496,981,290,961

So, the term with the greatest coefficient in the expansion of (2 + 3x)^9 when x = 3/2 is 496,981,290,961.