Chapter 7.5, Problem 74MR

To determine

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

Answer to Problem 74MR

The pattern of the sequence $\langle s_n\rangle =\frac{1}{2},\frac{1}{3},\frac{1}{4},\dots$ is $s_n=\frac{1}{n+1},n=1,2,3,\dots$ .., and next three terms of the sequence are $\frac{1}{5},\frac{1}{6},$ and $\frac{1}{7}$ .

Explanation of Solution

Given information:

The given sequence is $\frac{1}{2},\frac{1}{3},\frac{1}{4},\dots$

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 $\langle s_n\rangle =\frac{1}{2},\frac{1}{3},\frac{1}{4},\dots$ .

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

  $\langle s_n\rangle =\frac{1}{2},\frac{1}{3},\frac{1}{4},\dots ,\frac{1}{n+1},\dots$ , where $n=1,2,3,\dots$

Thus, the formula or the pattern of the sequence $\langle s_n\rangle$ is $s_n=\frac{1}{n+1},n=1,2,3,\dots$ .

From the formula, $s_n=\frac{1}{n+1},n=1,2,3,\dots$ the sequence may be written as:

  $\langle s_n\rangle =\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},\frac{1}{6},\frac{1}{7},\dots ,\frac{1}{n+1},\dots$

Hence, next three terms of the sequence $\langle s_n\rangle =\frac{1}{2},\frac{1}{3},\frac{1}{4},\dots$ are $\frac{1}{5},\frac{1}{6},\frac{1}{7}$ .

Conclusion:

The pattern of the sequence $\langle s_n\rangle =\frac{1}{2},\frac{1}{3},\frac{1}{4},\dots$ is $s_n=\frac{1}{n+1},n=1,2,3,\dots$ .., and next three terms of the sequence are $\frac{1}{5},\frac{1}{6},$ and $\frac{1}{7}$ .