Answer :
The sum of the middle three terms in the arithmetic progression is approximately [tex]\(10.5\)[/tex], closest to option C, which is [tex]\(11\)[/tex].
Let's denote the common difference of the arithmetic progression (AP) as [tex]\(d\)[/tex] and the first term as [tex]\(a\).[/tex] The sum of the first 3 terms [tex](\(S_3\))[/tex] is given by the formula [tex]\(S_3 = \frac{3}{2}[2a + 2d]\)[/tex], and the sum of the last 3 terms [tex](\(S_{11}\))[/tex] is given by [tex]\(S_{11} = \frac{3}{2}[2a + 8d]\).[/tex]
Given that [tex]\(S_3 = 6\)[/tex] and [tex]\(S_{11} = 16\)[/tex], we can set up the following equations:
[tex]\[3a + 3d = 6 \quad \text{(equation 1)}\][/tex]
[tex]\[3a + 12d = 16 \quad \text{(equation 2)}\][/tex]
Now, solving these simultaneous equations, we find [tex]\(a = 2\)[/tex] and [tex]\(d = 0.5\).[/tex]The common difference is [tex]\(0.5\)[/tex] and the first term is [tex]\(2\).[/tex]
The sum of the middle three terms [tex](\(S_7\))[/tex] can be expressed as [tex]\(S_7 = 3a + 9d\)[/tex]. Substituting the values, we get [tex]\(S_7 = 3(2) + 9(0.5) = 6 + 4.5 = 10.5\)[/tex].
Therefore, the correct answer is not provided in the options given. However, the closest value to [tex]\(10.5\) is \(11\).[/tex]
