Let's determine how many baseball cards Jackson will have in his collection after 7 years by analyzing the pattern in the given data:
1. Identify the increments in years and cards:
- From year 2 to year 3, the number of cards increased from 26 to 39.
- From year 3 to year 5, the number of cards increased from 39 to 65.
2. Calculate the rate of increase per year:
- First interval (from year 2 to year 3):
[tex]\[
\text{Rate} = \frac{39 - 26}{3 - 2} = \frac{13}{1} = 13 \text{ cards per year}
\][/tex]
- Second interval (from year 3 to year 5):
[tex]\[
\text{Rate} = \frac{65 - 39}{5 - 3} = \frac{26}{2} = 13 \text{ cards per year}
\][/tex]
Since the rate remains consistent at 13 cards per year across the intervals we calculated, we can use this rate to predict future values.
3. Predict the number of cards after 7 years:
- We know the starting point is 26 cards at year 2.
- The additional number of cards from year 2 to year 7 is calculated using the consistent rate:
[tex]\[
\text{Additional years} = 7 - 2 = 5 \text{ years}
\][/tex]
[tex]\[
\text{Additional cards} = 5 \times 13 = 65 \text{ cards}
\][/tex]
4. Determine the total number of cards at year 7:
- Adding the additional cards to the initial amount:
[tex]\[
\text{Total cards} = 26 + 65 = 91 \text{ cards}
\][/tex]
Therefore, Jackson will have 91 cards after 7 years.
