Chapter 7.5, Problem 73MR

To determine

Todescribe:The pattern for given sequence and to write next three numbers in the sequence.

Answer to Problem 73MR

The pattern of the sequence $\langle s_n\rangle =10,4,-2,\dots$ is $s_{n+1}=s_n-6$ , and $s_1=10$ , $n=1,2,3,\dots$ .and next three terms of the sequence are -8, -14, and -20.

Explanation of Solution

Given information:

The given sequence is $10,4,-2,\dots$

Concept used:

A sequence is an arrangement of a list of numbers in a definite order. The numbers forming the sequence are known as its terms. The definite order or the pattern of the sequence

can be expressed by a formula involving natural number n .

Calculations:

The given sequence is $\langle s_n\rangle =10,4,-2,\dots$ .

The consecutive terms of the sequence are such that a term is 6 less than preceding term. Therefore, the sequence $\langle s_n\rangle$ may be written as

  $\langle s_n\rangle =10,(10-6),(4-6),\dots ,s_n=\left(s_{n-1}-6\right),s_{n+1}=\left(s_n-6\right)$

Thus, the formula or the pattern of the sequence $\langle s_n\rangle$ is $s_{n+1}=s_n-6$ , and $s_1=10$ , $n=1,2,3,\dots$ .

From the formula, $s_{n+1}=s_n-6$ the sequence may be written as:

  $\langle s_n\rangle =10,4,-2,\left(-2-6\right),\left(-8-6\right),\left(-14-6\right),\dots$

  $\Rightarrow \langle s_n\rangle =10,4,-2,-8,-14,-20,\dots$

Hence, next three terms of the sequence $\langle s_n\rangle =10,4,-2,\dots$ are -8, -14, and -20.

Conclusion:

The pattern of the sequence $\langle s_n\rangle =10,4,-2,\dots$ is $s_{n+1}=s_n-6$ , and $s_1=10$ , $n=1,2,3,\dots$ .and next three terms of the sequence are -8, -14, and -20.