Questions: Write an explicit formula for (an), the (n^th) term of the sequence (13,20,27, ldots)

Write an explicit formula for (an), the (n^th) term of the sequence (13,20,27, ldots)
Transcript text: Write an explicit formula for $a_{n}$, the $n^{\text {th }}$ term of the sequence $13,20,27, \ldots$
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Solution

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Solution Steps

Step 1: Identify the Sequence

The given sequence is \( 13, 20, 27, \ldots \). This is an arithmetic sequence where the first term \( a_1 = 13 \).

Step 2: Determine the Common Difference

To find the common difference \( d \), we calculate: \[ d = 20 - 13 = 7 \]

Step 3: Write the Explicit Formula

The explicit formula for the \( n^{\text{th}} \) term of an arithmetic sequence is given by: \[ a_n = a_1 + (n - 1) \cdot d \] Substituting the values of \( a_1 \) and \( d \): \[ a_n = 13 + (n - 1) \cdot 7 \]

Step 4: Calculate the 5th Term

To find the 5th term \( a_5 \): \[ a_5 = 13 + (5 - 1) \cdot 7 = 13 + 4 \cdot 7 = 13 + 28 = 41 \]

Final Answer

The 5th term of the sequence is \( \boxed{41} \).

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