The first five terms of a sequence are "a" Subscript, 1 , Baseline equals 47 , "a" Subscript, 2 , Baseline equals 33 , "a" Subscript, 3 , Baseline equals 19 , "a" Subscript, 4 , Baseline equals 5 , "a" Subscript, 5 , Baseline equals negative 9.a1=47, a2=33, a3=19, a4=5, a5=−9.

Question

The first five terms of a sequence are  "a" Subscript, 1 , Baseline equals 47 , "a" Subscript, 2 , Baseline equals 33 , "a" Subscript, 3 , Baseline equals 19 , "a" Subscript, 4 , Baseline equals 5 , "a" Subscript, 5 , Baseline equals negative 9.a1​=47, a2​=33, a3​=19, a4​=5, a5​=−9.

Based on this information, create an equation that can be used to find the nth term of the sequence, "a" Subscript, "n" , Baselinean​.

GPT-4o mini · Answer

To find a general formula for the $ n $-th term of the sequence given: - $ a_1 = 47 $ - $ a_2 = 33 $ - $ a_3 = 19 $ - $ a_4 = 5 $ - $ a_5 = -9 $ we first observe the differences between consecutive terms. 1. $ a_2 - a_1 = 33 - 47 = -14 $ 2. $ a_3 - a_2 = 19 - 33 = -14 $ 3. $ a_4 - a_3 = 5 - 19 = -14 $ 4. $ a_5 - a_4 = -9 - 5 = -14 $ The sequence has a constant difference of $-14$, indicating that it is an arithmetic sequence. The general form for the $ n $-th term of an arithmetic sequence can be expressed as: $$ a_n = a_1 + (n - 1) \cdot d $$ where: - $ a_1 $ is the first term of the sequence, - $ d $ is the common difference, - $ n $ is the term number. Substituting the values we have: - $ a_1 = 47 $ - $ d = -14 $ Thus, the formula becomes: $$ a_n = 47 + (n - 1)(-14) $$ We can simplify this expression: $$ a_n = 47 - 14(n - 1) $$ $$ a_n = 47 - 14n + 14 $$ $$ a_n = 61 - 14n $$ Therefore, the equation to find the $ n $-th term of the sequence is: $$ a_n = 61 - 14n $$

The first five terms of a sequence are "a" Subscript, 1 , Baseline equals 47 , "a" Subscript, 2 , Baseline equals 33 , "a" Subscript, 3 , Baseline equals 19 , "a" Subscript, 4 , Baseline equals 5 , "a" Subscript, 5 , Baseline equals negative 9.a1​=47, a2​=33, a3​=19, a4​=5, a5​=−9.

The first five terms of a sequence are "a" Subscript, 1 , Baseline equals 47 , "a" Subscript, 2 , Baseline equals 33 , "a" Subscript, 3 , Baseline equals 19 , "a" Subscript, 4 , Baseline equals 5 , "a" Subscript, 5 , Baseline equals negative 9.a1=47, a2=33, a3=19, a4=5, a5=−9.