<PRE><B>Question 935346</B>

Find the coefficient of x^4 in the binomial expansion of (5+2x)^7
(with steps please)
&lt;pre&gt;&lt;font face = &quot;Tohoma&quot; size = 4 color = &quot;indigo&quot;&gt;&lt;b&gt;Formula for a specific term in an expansion: (a + b)&lt;sup&gt;n&lt;/sup&gt; = &lt;sub&gt;n&lt;/sub&gt;C&lt;sub&gt;r-1&lt;/sub&gt;(a)&lt;sup&gt;n-(r-1)&lt;/sup&gt;(b)&lt;sup&gt;r-1&lt;/sup&gt;, where r = term number 
(5 + 2x)&lt;sup&gt;7&lt;/sup&gt;
Observing the binomial, it’s seen that the x-term, or “b” in the expansion of the binomial needs to be to the 4&lt;sup&gt;th&lt;/sup&gt; power. 
Thus, the term that produces {{{b^4}}} is the term that needs to be determined. Thus, we can say that: {{{b^4 = b^(r - 1)}}}, and that: 
4 = r – 1
4 + 1 = r
r, or term number = 5

Using the formula for a specific term in an expansion, term 5 or the 5th term can be determined, as follows: 
 (a + b)&lt;sup&gt;n&lt;/sup&gt; = &lt;sub&gt;n&lt;/sub&gt;C&lt;sub&gt;r-1&lt;/sub&gt;(a)&lt;sup&gt;n-(r-1)&lt;/sup&gt;(b)&lt;sup&gt;r-1&lt;/sup&gt;, where r = term number 
(5 + 2x)&lt;sup&gt;7&lt;/sup&gt; = &lt;sub&gt;7&lt;/sub&gt;C&lt;sub&gt;5-1&lt;/sub&gt;(5)&lt;sup&gt;7-(5-1)&lt;/sup&gt;(2x)&lt;sup&gt;5-1&lt;/sup&gt;, where r = 5&lt;sup&gt;th&lt;/sup&gt; term 
(5 + 2x)&lt;sup&gt;7&lt;/sup&gt; = &lt;sub&gt;7&lt;/sub&gt;C&lt;sub&gt;4&lt;/sub&gt;(5)&lt;sup&gt;7-4&lt;/sup&gt;(2x)&lt;sup&gt;4&lt;/sup&gt;
(5 + 2x)&lt;sup&gt;7&lt;/sup&gt; = 35(5)&lt;sup&gt;3&lt;/sup&gt;(2x)&lt;sup&gt;4&lt;/sup&gt;
(5 + 2x)&lt;sup&gt;7&lt;/sup&gt; = 4,375(16)x&lt;sup&gt;4&lt;/sup&gt;
(5 + 2x)&lt;sup&gt;7&lt;/sup&gt; = 70,000x&lt;sup&gt;4&lt;/sup&gt;
Thus, the {{{x^4}}} term, or 5th term of the binomial expansion: {{{(5 + 2x)^7}}}, is 70,000x&lt;sup&gt;4&lt;/sup&gt;, which means
that the coefficient is {{{highlight_green(highlight_green(70000))}}}

Not to be confused, if the binomial&#39;s expressions were rearranged from {{{(5 + 2x)^7}}} to {{{(2x + 5)^7}}},
then {{{a^4}}} or {{{(2x)^4}}} would be the 4&lt;sup&gt;th&lt;/sup&gt; term.
You can do the check!! 
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