Question Write an explicit formula for \( a_{n} \), the \( n^{\text {th }} \) term of the sequence \( 36,44,52, \ldots \) Answer Attempt 1 out of 3 \( a_{n}= \)

To find the explicit formula for the sequence \( 36, 44, 52, \ldots \), notice that this sequence is an arithmetic sequence where the first term \( a_1 = 36 \) and the common difference \( d = 8 \) (since \( 44 - 36 = 8 \) and \( 52 - 44 = 8 \)). The \( n^{\text{th}} \) term of an arithmetic sequence can be expressed as: \[ a_{n} = a_1 + (n-1)d \] Substituting the known values gives: \[ a_{n} = 36 + (n-1) \cdot 8 \] Simplifying this, we get: \[ a_{n} = 36 + 8n - 8 = 8n + 28 \] So the explicit formula for the \( n^{\text{th}} \) term is: \[ a_{n} = 8n + 28 \] Can you imagine students in ancient Greece brainstorming sequences while discussing harmony in music? They would marvel at how patterns in numbers can create beautiful sounds, just as they saw in their philosophies! In the modern world, arithmetic sequences like the one we've explored are all around us — from calculating expenses over time to tracking growth rates in finance. Teachers and managers alike appreciate the clarity these sequences provide when predicting future outcomes, making them an essential tool for decision-making!