To find the general term in the expansion of (2x + 3)^5, we use the binomial theorem formula for the general term T_r+1 of (a+b)^n:
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To find the general term in the expansion of $(2x + 3)^5$, we use the binomial theorem formula for the general term $T_{r+1}$ of $(a+b)^n$:
$$T_{r+1} = \binom{n}{r} a^{n-r} b^r$$
In this expansion:
• $a = 2x$
• $b = 3$
• $n = 5$
Substitute these values into the formula:
$$T_{r+1} = \binom{5}{r} (2x)^{5-r} (3)^r$$
This expression represents the general term. Let's compare it with the given options.
The options are:
(a) $\binom{5}{r} (2x)^r$
(b) $\binom{5}{r} (2x)^{5-r} . 3^r$
(c) $\binom{5}{r} 2^r . 3^{5-r}$
(d) $\binom{5}{r} x^r . 3^{5-r}$
Our derived general term matches option (b).
The final answer is $\boxed{\text{(b)}}$.
