Chapter SH, Problem 7.35EP

To determine

To find first five terms of the sequence.

Answer to Problem 7.35EP

First five terms are

  $12,-\frac{7}{2},\frac{17}{4},\frac{3}{8},\frac{1}{2}$

Explanation of Solution

Given information:

  $a_1=12$

The sequence is

  $a_n=-\frac{1}{2}{a}_{n-1}+\frac{5}{2},n\ge 2$

Calculation:

  $a_n=-\frac{1}{2}{a}_{n-1}+\frac{5}{2},n\ge 2\mathrm{......}(1)$

  $a_n$ is $n^{th}$ term

Put $n=2$in equation (1)

  $a_2=-\frac{1}{2}{a}_1+\frac{5}{2}$

On solving,

  $a_2=-\frac{1}{2}(12)+\frac{5}{2}$

On solving,

  $a_2=-\frac{7}{2}$

Put $n=3$in equation (1)

  $a_3=-\frac{1}{2}{a}_2+\frac{5}{2}$

On calculation $a_2=-\frac{7}{2}$ put in above

  $a_3=-\frac{1}{2}\left(-\frac{7}{2}\right)+\frac{5}{2}$

On solving,

  $a_3=\frac{17}{4}$

Put $n=4$in equation (1)

  $a_4=-\frac{1}{2}{a}_3+\frac{5}{2}$

On calculation $a_3=\frac{17}{4}$ put in above

  $a_4=-\frac{1}{2}\left(\frac{17}{4}\right)+\frac{5}{2}$

On solving,

  $a_4=\frac{3}{8}$

Put $n=5$in equation (1)

  $a_5=-\frac{1}{2}{a}_4+\frac{5}{2}$

On calculation $a_4=\frac{3}{8}$ put in above

  $a_5=-\frac{1}{2}\left(\frac{3}{8}\right)+\frac{5}{2}$

On solving,

  $a_5=\frac{1}{2}$

Hence, first five terms are

  $12,-\frac{7}{2},\frac{17}{4},\frac{3}{8},\frac{1}{2}$