Question

Enzo’s profit over the last few months has been falling. In December he made $900, in January his profit was $500, and in March, he lost $300. Is the sequence that represents his profit arithmetic? If it is, what is the recursive formula for the sequence?

Answer 1

See explanation

Explanation:

Enzo’s profit over the last few months:

• In December, he made profit $900;
• In January, his his profit was $500;
• In February - unknown;
• In March, his loss was $300.

Find the difference between given data

Dec - Jan
`\$900-\$500=\$400`

Jan - Feb
`\$500-\$x`

Feb - Mar
`\$x-(-\$300)=\$x+\$300`

If this sequence is arithmetic, then

`\$500-\$x=\$x+\$300=\$400\\ \\\$x=\$100`

So, if the profit in February was $100, then this sequence is arithmetic sequence. In this case,

`a_1=900\\ \\a_2=500\\ \\a_3=100\\ \\a_4=-300\\ \\...\\ \\d=-400`

The recursive formula is

`a_n=a_(n-1)-400`