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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:

Mathematics

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d

(b)\text{(b)}

To find the general term in the expansion of (2x+3)5(2x + 3)^5, we use the binomial theorem formula for the general term Tr+1T_{r+1} of (a+b)n(a+b)^n:

Tr+1=(nr)an−rbrT_{r+1} = \binom{n}{r} a^{n-r} b^r

In this expansion:

  • a=2xa = 2x
  • b=3b = 3
  • n=5n = 5

Substitute these values into the formula:

Tr+1=(5r)(2x)5−r(3)rT_{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) (5r)(2x)r\binom{5}{r} (2x)^r

(b) (5r)(2x)5−r.3r\binom{5}{r} (2x)^{5-r} . 3^r

(c) (5r)2r.35−r\binom{5}{r} 2^r . 3^{5-r}

(d) (5r)xr.35−r\binom{5}{r} x^r . 3^{5-r}

Our derived general term matches option (b).

The final answer is (b)\boxed{\text{(b)}}.

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