For a fractional form a_{n+1} = (pa_n + q)/(ra_n + s), replacing it with the reciprocal b_n = 1/a_n may make it linear.
For a form containing n, a_{n+1} = pa_n + f(n), dividing both sides by pⁿ⁺¹ produces a difference form.
Example ① Find the general term for a_1=1, a_{n+1}=2a_n+n.
Solution: Divide both sides by 2ⁿ⁺¹: a_{n+1}/2ⁿ⁺¹ = a_n/2ⁿ + n/2ⁿ⁺¹
Let b_n = a_n/2ⁿ. This becomes a difference form → b_{n+1}−b_n = n/2ⁿ⁺¹.
(The calculation is complex; the final form is of the form a_n = C·2ⁿ − n − 1.)
Solution: Divide both sides by 2ⁿ⁺¹: a_{n+1}/2ⁿ⁺¹ = a_n/2ⁿ + n/2ⁿ⁺¹
Let b_n = a_n/2ⁿ. This becomes a difference form → b_{n+1}−b_n = n/2ⁿ⁺¹.
(The calculation is complex; the final form is of the form a_n = C·2ⁿ − n − 1.)
Key Point
- Learn the substitution technique for each pattern
Practice Problems
- For a_1=1, a_{n+1}=a_n/(a_n+1), find the recurrence relation for the reciprocal b_n=1/a_n.
Solutions and Explanations
- Solution:b_{n+1} = b_n + 1 → arithmetic sequence → b_n=n, a_n=1/n.