Write a recursive formula for the sequence 7,4,1,2,5,.... , then find the next term. (a).an=an...


Write a recursive formula for the sequence {eq}7, 4, 1, 2, 5, .... {/eq}, then find the next term.

(a).{eq}a_n = a_{n-1}- 3 {/eq}, where {eq}a1 = 8; 7 {/eq}

(b). {eq}a_n = a_{n-1}- 3 {/eq}, where {eq}a_1 = 7; 8 {/eq}

(c). {eq}a_n = a_{n-1} + 3 {/eq}, where {eq}a_1 = 3; 22 {/eq}

(d). {eq}a_n = a_{n-1} + 3 {/eq}, where {eq}a_1 = 7; 8 {/eq}

Arithmetic Sequence

An arithmetic sequence is a type of sequence consisting of array of terms in which the next term of the sequence is obtained by adding a common difference to the preceding term. The general formula for an arithmetic sequence takes the form {eq}a_n = a_1 + (n-1)d {/eq}, where

  • {eq}a_n {/eq} is the {eq}nth {/eq} term of the sequence,
  • {eq}a_1 {/eq} is the first term,
  • {eq}n {/eq} is the number of terms; and,
  • {eq}d {/eq} is called the common difference.

Answer and Explanation: 1

I suppose there is a mistake in the question itself. The sequence should be {eq}7,4,1,-2,-5,... {/eq} for it to be an arithmetic sequence and for us to arrive at a recursive formula. Here, we can see that our first term is {eq}a_1 = 7 {/eq} with a common difference {eq}d = -3 {/eq}. By substitution, the recursive formula is given as

$$\begin{align} a_n &= a_1 + (n-1)d\\[0.3cm] a_n &= 7 + (n-1)-3\\[0.3cm] a_n &= 7 -3n+ 3\\[0.3cm] a_n &= 10 - 3n\\[0.3cm] \end{align} $$

The next term is obtained by adding {eq}-3 {/eq} to {eq}-5 {/eq}. Hence, the next term on the sequence is {eq}-8 {/eq}. None on the given choices makes sense.

Learn more about this topic:

Arithmetic Sequence: Formula & Definition


Chapter 25 / Lesson 3

Discover the arithmetic sequence definition and how math uses it. Know its formula and how to solve problems relating to it through sample calculations.

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