- Calculate terms by substituting \(n=1\) through \(5\) into the expressions to find the first five terms of each sequence.
- Find specific sequence terms by substituting the given position number \(n\) directly into the expression.
- Sequences: \(3n+1 \rightarrow 4, 7, 10, 13, 16\); \(4n \rightarrow 4, 8, 12, 16, 20\); \(2n-1 \rightarrow 1, 3, 5, 7, 9\); \(5n+2 \rightarrow 7, 12, 17, 22, 27\).
- Specific terms: 10th of \(2n+1\) is \(\boxed{21}\), 8th of \(6n+3\) is \(\boxed{51}\), 5th of \(4n+1\) is \(\boxed{21}\).
Explanation
- Understanding Sequence Calculation
The objective is to find the first five terms of four different algebraic sequences and to find specific terms for three other sequences. An algebraic expression for a sequence allows us to calculate any term by substituting the position number, denoted by \(n\), into the expression. For the first part, we calculate terms for \(n=1, 2, 3, 4, 5\). For the second part, we substitute specific values of \(n\) to find the requested terms.
- Calculating the First Five Terms
To find the first five terms of each sequence, we substitute \(n = 1, 2, 3, 4, 5\) into the expressions:
- For \(3n+1\):
- \(n=1: 3(1)+1 = 4\)
- \(n=2: 3(2)+1 = 7\)
- \(n=3: 3(3)+1 = 10\)
- \(n=4: 3(4)+1 = 13\)
- \(n=5: 3(5)+1 = 16\)
- For \(4n\):
- \(n=1: 4(1) = 4\)
- \(n=2: 4(2) = 8\)
- \(n=3: 4(3) = 12\)
- \(n=4: 4(4) = 16\)
- \(n=5: 4(5) = 20\)
- For \(2n-1\):
- \(n=1: 2(1)-1 = 1\)
- \(n=2: 2(2)-1 = 3\)
- \(n=3: 2(3)-1 = 5\)
- \(n=4: 2(4)-1 = 7\)
- \(n=5: 2(5)-1 = 9\)
- For \(5n+2\):
- \(n=1: 5(1)+2 = 7\)
- \(n=2: 5(2)+2 = 12\)
- \(n=3: 5(3)+2 = 17\)
- \(n=4: 5(4)+2 = 22\)
- \(n=5: 5(5)+2 = 27\)
- Finding Specific Sequence Terms
Now we find the specific terms requested by substituting the given \(n\) values:
- The 10th term of \(2n+1\):
- Substitute \(n=10\): \(2(10)+1 = 20+1 = 21\)
- The 8th term of \(6n+3\):
- Substitute \(n=8\): \(6(8)+3 = 48+3 = 51\)
- The 5th term of \(4n+1\):
- Substitute \(n=5\): \(4(5)+1 = 20+1 = 21\)
- Conclusion
In summary, we have calculated the sequences and specific terms as follows:
- The first five terms for \(3n+1\) are \(4, 7, 10, 13, 16\).
- The first five terms for \(4n\) are \(4, 8, 12, 16, 20\).
- The first five terms for \(2n-1\) are \(1, 3, 5, 7, 9\).
- The first five terms for \(5n+2\) are \(7, 12, 17, 22, 27\).
- The 10th term of \(2n+1\) is \(21\).
- The 8th term of \(6n+3\) is \(51\).
- The 5th term of \(4n+1\) is \(21\).
Examples
Algebraic sequences are used in many real-life situations, such as calculating savings or planning projects. For example, if you start a piggy bank with \$2 and add \$3 each week, the amount in the bank after \(n\) weeks can be represented by the sequence \(3n+2\). After 5 weeks, you would have \(3(5)+2 = \$17\). Understanding these patterns helps you predict outcomes and manage resources effectively.