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Magazine Chapter 6: Problem 4
In \(3-18,\) write the first five terms of each sequence. $$ a_{n}=n+5 $$
Short Answer
Expert verified
The first five terms are 6, 7, 8, 9, and 10.
Step by step solution
01
Understand the Sequence Formula
The sequence is defined as \( a_n = n + 5 \). This formula gives us the general term of the sequence based on the index \( n \). To find the specific terms, we will substitute the values of \( n = 1, 2, 3, 4, \) and \( 5 \) into the formula.
02
Calculate the First Term
To find the first term, substitute \( n = 1 \) into the formula:\[ a_1 = 1 + 5 = 6. \]Thus, the first term is 6.
03
Calculate the Second Term
Substitute \( n = 2 \) into the formula:\[ a_2 = 2 + 5 = 7. \]Therefore, the second term is 7.
04
Calculate the Third Term
Substitute \( n = 3 \) into the formula:\[ a_3 = 3 + 5 = 8. \]So, the third term is 8.
05
Calculate the Fourth Term
Substitute \( n = 4 \) into the formula:\[ a_4 = 4 + 5 = 9. \]Hence, the fourth term is 9.
06
Calculate the Fifth Term
Substitute \( n = 5 \) into the formula:\[ a_5 = 5 + 5 = 10. \]Thus, the fifth term is 10.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Sequence Formula
An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This difference is known as the common difference. To find any term in an arithmetic sequence, we use a sequence formula. The general sequence formula for an arithmetic sequence can be written as follows:
- \( a_n = a_1 + (n-1) \cdot d \)
Terms Calculation
Terms calculation involves substituting the position of each term into the sequence formula to find the corresponding term value. In our exercise, the formula is given as \( a_n = n + 5 \). To calculate the terms:
- First term: Substitute \( n = 1 \), resulting in \( a_1 = 1 + 5 = 6 \).
- Second term: Substitute \( n = 2 \), resulting in \( a_2 = 2 + 5 = 7 \).
- Third term: Substitute \( n = 3 \), leading to \( a_3 = 3 + 5 = 8 \).
- Fourth term: Substitute \( n = 4 \), resulting in \( a_4 = 4 + 5 = 9 \).
- Fifth term: Substitute \( n = 5 \), leading to \( a_5 = 5 + 5 = 10 \).
Index Substitution
Index substitution is a crucial step in finding the specific terms of a sequence. It's the process of substituting specific values of the index \( n \) into the sequence formula to determine the sequence's term at that position. For example, in the sequence formula \( a_n = n + 5 \), trying different values of \( n \) effectively gives us each term:
- For \( n = 1 \), substituting gives \( a_1 = 1 + 5 = 6 \).
- For \( n = 2 \), substituting gives \( a_2 = 2 + 5 = 7 \).
- Continuing this process similarly helps us find further terms.
Explicit Formula
An explicit formula in sequences is a direct way to find any term without knowing the previous term. The explicit formula gives a direct expression where the term can be calculated explicitly for any position in the sequence. In arithmetic sequences, the explicit formula is:
- \( a_n = a_1 + (n-1) \cdot d \)
