What is the fourth term in the binomial expansion of (2x + 5)^5?

To find the fourth term in the expansion of the binomial expression (2x + 5)⁵, we can use the Binomial Theorem. The Binomial Theorem states that:

• (a + b)ⁿ = from k = 0 to n
  C(n, k) a^n-k b^k,

where C(n, k) is the binomial coefficient, calculated as C(n, k) = n! / (k!(n-k)!).

In our case, we have a = 2x, b = 5, and n = 5.

We need to find the fourth term, which corresponds to k = 3 (since the first term corresponds to k = 0). Therefore, we can plug these values into the formula:

Term(k+1) = C(n, k) a^n-k b^k.

Substituting our values:

• C(5, 3) = 5! / (3!(5-3)!) = 10.
• a^n-k = (2x)^5-3 = (2x)² = 4x².
• b^k = 5³ = 125.

Now, we can put it all together:

Fourth Term = C(5, 3) * (2x)² * 5³

Fourth Term = 10 * 4x² * 125

Fourth Term = 5000 x².

Thus, the fourth term in the expansion of (2x + 5)⁵ is 5000x².