To solve for the correct sigma notation of the given series:
[tex]\[ 2.5 + 2.5(1.2) + 2.5(1.2)^2 + \cdots + 2.5(1.2)^{87} \][/tex]
we first need to understand the general term in the series. Observing the pattern, each term in the series can be expressed as:
[tex]\[ 2.5 \cdot (1.2)^k \][/tex]
where [tex]\( k \)[/tex] starts from [tex]\( 0 \)[/tex] and goes up to [tex]\( 87 \)[/tex]. Therefore, in mathematical notation, we have:
[tex]\[ \sum_{k=0}^{87} 2.5 \cdot (1.2)^k \][/tex]
However, we need to adjust our notation to match the options given in the problem. The options all start the summation from [tex]\( k = 1 \)[/tex].
To convert the exponent in our series to match [tex]\( k = 1 \)[/tex] to [tex]\( 87 \)[/tex] or [tex]\( 88 \)[/tex], we must adjust the exponent.
Given:
[tex]\[ \sum_{k=0}^{87} 2.5 \cdot (1.2)^k \][/tex]
To adjust for the range [tex]\( k=1 \)[/tex] to [tex]\( 87 \)[/tex], note that:
[tex]\[ \sum_{k=1}^{87} 2.5 \cdot (1.2)^{k-1} \][/tex]
Shifting the exponent [tex]\( k \)[/tex] by [tex]\( 1 \)[/tex] backward (i.e., for [tex]\( k = 1 \)[/tex], exponent is [tex]\( 0 \)[/tex]; for [tex]\( k = 2 \)[/tex], exponent is [tex]\( 1 \)[/tex]; etc.) maintains the correct series pattern.
Hence, the correct sigma notation that matches the given series is:
[tex]\[ \sum_{k=1}^{87} 2.5 \cdot (1.2)^{k-1} \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{\text{B}} \][/tex]
