Question

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the first 5 terms of an arithmetic sequence are 3, 15, 27, 39 and 51.
let ( t_n ) represent the ( n )th term of the sequence. what is the value of ( t_{13} )?

Explanation:

Step1: Find the common difference \( d \)

In an arithmetic sequence, the common difference \( d \) is the difference between consecutive terms. Let's take the first two terms: \( d = 15 - 3 = 12 \). We can check with other terms, e.g., \( 27 - 15 = 12 \), so \( d = 12 \).

Step2: Recall the formula for the \( n \)-th term of an arithmetic sequence

The formula for the \( n \)-th term of an arithmetic sequence is \( t_n = t_1 + (n - 1)d \), where \( t_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference. Here, \( t_1 = 3 \), \( d = 12 \), and we need to find \( t_{13} \), so \( n = 13 \).

Step3: Substitute the values into the formula

Substitute \( t_1 = 3 \), \( d = 12 \), and \( n = 13 \) into the formula: \( t_{13} = 3 + (13 - 1) \times 12 \). First, calculate \( 13 - 1 = 12 \). Then, \( 12 \times 12 = 144 \). Finally, \( 3 + 144 = 147 \).

Answer:

\( 147 \)