To match each series with its corresponding sigma notation, let's identify the patterns in each series and compare them with the provided sigma notations.
1. Series: 2 + 6 + 18 + 54 + 162
- This series is a geometric progression where each term is obtained by multiplying the previous term by 3.
- The first term [tex]\(a\)[/tex] is 2.
- Common ratio [tex]\(r\)[/tex] is 3.
- The general term of a geometric series can be written as [tex]\(a \cdot r^n = 2 \cdot 3^n\)[/tex].
- Summing from [tex]\(n = 0\)[/tex] to 4: [tex]\(\sum_{n=0}^4 2(3)^n\)[/tex].
2. Series: 3 + 15 + 75 + 375 + 1,875
- This series is a geometric progression where each term is obtained by multiplying the previous term by 5.
- The first term [tex]\(a\)[/tex] is 3.
- Common ratio [tex]\(r\)[/tex] is 5.
- The general term of the series can be written as [tex]\(a \cdot r^n = 3 \cdot 5^n\)[/tex].
- Summing from [tex]\(n = 0\)[/tex] to 4: [tex]\(\sum_{n=0}^4 3(5)^n\)[/tex].
3. Series: 3 + 12 + 48 + 192 + 768
- This series is a geometric progression where each term is obtained by multiplying the previous term by 4.
- The first term [tex]\(a\)[/tex] is 3.
- Common ratio [tex]\(r\)[/tex] is 4.
- The general term of the series can be written as [tex]\(a \cdot r^n = 3 \cdot 4^n\)[/tex].
- Summing from [tex]\(n = 0\)[/tex] to 4: [tex]\(\sum_{n=0}^4 3(4)^n\)[/tex].
4. Series: 4 + 32 + 256 + 2,048 + 16,384
- This series is a geometric progression where each term is obtained by multiplying the previous term by 8.
- The first term [tex]\(a\)[/tex] is 4.
- Common ratio [tex]\(r\)[/tex] is 8.
- The general term of the series can be written as [tex]\(a \cdot r^n = 4 \cdot 8^n\)[/tex].
- Summing from [tex]\(n = 0\)[/tex] to 4: [tex]\(\sum_{n=0}^4 4(8)^n\)[/tex].
Given these observations, we match the series with their corresponding sigma notations as follows:
- [tex]\(2 + 6 + 18 + 54 + 162\)[/tex] corresponds to [tex]\(\sum_{n=0}^4 2(3)^n\)[/tex].
- [tex]\(3 + 15 + 75 + 375 + 1,875\)[/tex] corresponds to [tex]\(\sum_{n=0}^4 3(5)^n\)[/tex].
- [tex]\(3 + 12 + 48 + 192 + 768\)[/tex] corresponds to [tex]\(\sum_{n=0}^4 3(4)^n\)[/tex].
- [tex]\(4 + 32 + 256 + 2,048 + 16,384\)[/tex] corresponds to [tex]\(\sum_{n=0}^4 4(8)^n\)[/tex].
So the final matches are:
1. [tex]\(2 + 6 + 18 + 54 + 162 \leftrightarrow \sum_{n=0}^4 2(3)^n\)[/tex]
2. [tex]\(3 + 15 + 75 + 375 + 1,875 \leftrightarrow \sum_{n=0}^4 3(5)^n\)[/tex]
3. [tex]\(3 + 12 + 48 + 192 + 768 \leftrightarrow \sum_{n=0}^4 3(4)^n\)[/tex]
4. [tex]\(4 + 32 + 256 + 2,048 + 16,384 \leftrightarrow \sum_{n=0}^4 4(8)^n\)[/tex]
