Hung saves 42 dollars in the first week and adds 8 dollars to the account every week after that. The guitar he wants costs 400 dollars. In which week does he first have enough?

Step-by-step solution

1. 1

   Write the balance as an arithmetic sequence. The amount in the account at the end of week $n$ is $u_n=42+(n-1)\cdot 8=8n+34$ with $u_1=42$ and $d=8$. This is a term of the sequence, not a sum, because the account carries its balance forward.

2. 2

   Set up the inequality. $8n+34\ge 400\quad\Longrightarrow\quad 8n\ge 366\quad\Longrightarrow\quad n\ge 45.75$

3. 3

   Round up to a whole week. Money only arrives at week boundaries, so the first admissible integer is $n=46$.

4. 4

   Check the two neighbouring weeks. $u_{45}=8(45)+34=394<400,\qquad u_{46}=8(46)+34=402\ge 400$

5. 5

   Answer. He can buy the guitar in week $46$, option D, with \2$ to spare.