Question

Find the next four terms of the following sequence: a_1=3a_n=2a_n-1+n

Answer 1

Given:

$\begin{gathered} \text{ A}_1\text{ = 3} \\ \text{ A}_n=2A_{n-1}\text{ + n} \end{gathered}$

Where,

n = term position

Since we already got the 1st term (A1 = 3), let's determine the next four terms.

2nd term: n = 2

$\text{ A}_2=2A_{2-1}\text{ + }2$$\text{ A}_2=2A_1\text{ + }2$$\text{ A}_2=2(3)\text{ + }2$$\text{ A}_2=6\text{ + }2$$\text{ A}_2=8$

3rd term: n = 3

$\text{ A}_3=2A_{3-1}\text{ + }3$$\text{ A}_3=2A_2\text{ + }3$$\text{ A}_3=2(8)_{}\text{ + }3$$\text{ A}_3=16_{}\text{ + }3$$\text{ A}_3=19$

4th term: n = 4

$\text{ A}_4=2A_{4-1}\text{ + }4$$\text{ A}_4=2A_3\text{ + }4$$\text{ A}_4=2(19)\text{ + }4$$\text{ A}_4=38\text{ + }4$$\text{ A}_4=42$

5th term: n = 5

$\text{ A}_5=2A_{5-1}\text{ + }5$$\text{ A}_5=2A_4\text{ + }5$$\text{ A}_5=2(42)\text{ + }5$$\text{ A}_5=84\text{ + }5$$\text{ A}_5=89$

Therefore, the next four terms of the following sequence are: 8, 19, 42, 89.