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Match each series with the equivalent series written in sigma notation.

[tex]\[
\begin{array}{c}
\sum_{n=0}^4 2(3)^n \quad \sum_{n=0}^4 3(4)^n \quad \sum_{n=0}^4 4(4)^n \quad \sum_{n=0}^4 3(2)^n \quad \sum_{n=0}^4 5(3)^n \\
\sum_{n=0}^4 4(8)^n \quad \sum_{n=0}^4 3(5)^n \quad \sum_{n=0}^4 2(9)^n \\
\end{array}
\][/tex]

[tex]\[
\begin{array}{c}
3 + 15 + 75 + 375 + 1,875 \longleftrightarrow \sum_{n=0}^4 3(5)^n \\
2 + 6 + 18 + 54 + 162 \longleftrightarrow \sum_{n=0}^4 2(3)^n \\
4 + 32 + 256 + 2,048 + 16,384 \longleftrightarrow \sum_{n=0}^4 4(8)^n \\
3 + 12 + 48 + 192 + 768 \longleftrightarrow \sum_{n=0}^4 3(4)^n \\
\end{array}
\][/tex]

Sagot :

GinnyAnsweridn
Let's analyze and match each series with its equivalent sigma notation expression.

Series: 3, 15, 75, 375, 1,875
- This series appears to be a geometric series with the first term [tex]\( a = 3 \)[/tex] and a common ratio [tex]\( r = 5 \)[/tex].
- The sigma notation for this series is [tex]\( \sum_{n=0}^4 3(5)^n \)[/tex].
- Match: [tex]\( 3 + 15 + 75 + 375 + 1,875 \longleftrightarrow \sum_{n=0}^4 3(5)^n \)[/tex].

Series: 2, 6, 18, 54, 162
- This series appears to be a geometric series with the first term [tex]\( a = 2 \)[/tex] and a common ratio [tex]\( r = 3 \)[/tex].
- The sigma notation for this series is [tex]\( \sum_{n=0}^4 2(3)^n \)[/tex].
- Match: [tex]\( 2 + 6 + 18 + 54 + 162 \longleftrightarrow \sum_{n=0}^4 2(3)^n \)[/tex].

Series: 4, 32, 256, 2,048, 16,384
- This series appears to be a geometric series with the first term [tex]\( a = 4 \)[/tex] and a common ratio [tex]\( r = 8 \)[/tex].
- The sigma notation for this series is [tex]\( \sum_{n=0}^4 4(8)^n \)[/tex].
- Match: [tex]\( 4 + 32 + 256 + 2,048 + 16,384 \longleftrightarrow \sum_{n=0}^4 4(8)^n \)[/tex].

Series: 3, 12, 48, 192, 768
- This series appears to be a geometric series with the first term [tex]\( a = 3 \)[/tex] and a common ratio [tex]\( r = 4 \)[/tex].
- The sigma notation for this series is [tex]\( \sum_{n=0}^4 3(4)^n \)[/tex].
- Match: [tex]\( 3 + 12 + 48 + 192 + 768 \longleftrightarrow \sum_{n=0}^4 3(4)^n \)[/tex].

Final matches:
- [tex]\( 3 + 15 + 75 + 375 + 1,875 \longleftrightarrow \sum_{n=0}^4 3(5)^n \)[/tex]
- [tex]\( 2 + 6 + 18 + 54 + 162 \longleftrightarrow \sum_{n=0}^4 2(3)^n \)[/tex]
- [tex]\( 4 + 32 + 256 + 2,048 + 16,384 \longleftrightarrow \sum_{n=0}^4 4(8)^n \)[/tex]
- [tex]\( 3 + 12 + 48 + 192 + 768 \longleftrightarrow \sum_{n=0}^4 3(4)^n \)[/tex]
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