# Recursive and Explicit Formulas for Arithmetic Linear Sequences

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Recursive and Explicit Formulas for Arithmetic (Linear) Sequences

An arithmetic sequence is a sequence with a constant increase or decrease also known as the constant difference In the sequence 10, 40, 70, 100, …. The constant difference between the terms is 30

A recursive formula for a sequence would be: a 1= first term in the sequence an = term you are trying to find an-1 = previous term in the sequence d = constant difference

In the sequence 10, 40, 70, 100, …. The constant difference between the terms is 30 The first term of the sequence is 10 The recursive formula for this sequence would be: **I plugged in 10 for the first term and 30 for the constant difference** Write the recursive formula of the sequence 4, 7, 10, 13, ….

In the sequence 4, 7, 10, 13, …. To find the 5 th term recursively, I plug it into the formula I just made: an = an-1 + 3 a 5 = a 5 -1 + 3 a 5 = 13 + 3 a 5 = 16 in words: 5 th term equals the 4 th term plus 3

Explicit Formulas A formula that allows you to find the nth term of the sequence by substituting known values in the expression. An explicit formula for sequence would be: an = a 1 + d( n - 1) a 1= first term in the sequence an = current term in the sequence d = constant difference n = term number

In the sequence 10, 40, 70, 100, …. The constant difference between the terms is 30 The first term of the sequence is 10 The explicit formula for this sequence would be: an = 10 + 30( n - 1) which simplifies to: an = -20 + 30 n **I plugged in 10 for the first term and 30 for the constant difference and distributed** Write the explicit formula of the sequence 4, 7, 10, 13, …. an = ___ + ___( n - 1) Simplify: an = ___ + ___n

In the sequence 4, 7, 10, 13, …. To find the 11 th term explicitly, I plug in the into the formula I just made: an = 1 + 3 n a 11 = 1 + 3(11) a 11 = 34 Find the 15 th term of the sequence using the formula: an = 1 + 3 n

Example Test Question The first row of theater has 15 seats in it. Each subsequent row has 3 more seats that the previous row. If the last row has 78 seats, how many rows are in theater? Step 1: Create an explicit formula: Step 2: Figure out what you are solving for: Step 3: Solve for the last row number: