# Arithmetic Sequences - accountant problem.

I need help with $part$ $c$ of the following problem:

An accountant has a salary scheme as outlined below.

STARTING SALARY £16,000. ANNUAL INCREASES OF £1,000 GUARANTEED THEREAFTER.

Let £$u_n$ denote the accountant's salary at the beginning of the the $n$ th year, after any increases which are then due have been added.

a) Write down the values of $u_1$, $u_2$, $u_3$ and $u_4$.

Ans: $u_1=16000$, $u_2=17000$, $u_3=18000$, $u_4=19000$

b) Find a formula for $u_n$

Ans: $u_n=1000n+15000$

c)Find the value of $n$ for which $u_{n+1}=1.04u_n$ (that is corresponding to an annual increase of 4%).

Ans: ?

• You know what $u_{n+1}$ is. You know what $u_n$ is. Substitute in the equation and solve for $n$.
– user228113
Feb 6, 2016 at 17:38

Replace $u_{n+1}$ and $u_{n}$ with their values $$1000(n+1)+15000 = 1.04 (1000n+15000)$$ Can you complete the calculation?

• Please note starting salary is $16000$ Feb 6, 2016 at 17:45
• @ArchisWelankar Actually at $u_1$ $n = 1$ but there is no salary increase, it's why $1000$ is subtracted... OP's b) being correct. Feb 6, 2016 at 17:55

Let salary be $1000n+15000$ so next year it will be $1000n+16000$ now we want $u_{n+1}=1.04u_n$ plug in the values and get $n=10$

• what value did you get for n?
– Alan
Feb 6, 2016 at 17:43
• I got $n=9$ ... Feb 6, 2016 at 17:49
• No the value should be n=10
– Alan
Feb 6, 2016 at 17:52