Let's analyze Sam's penny collection step-by-step to determine the correct equation that defines the number of pennies he has after the nth day.
Day 1:
- Sam is given 10 pennies.
- After the first day, he has 10 pennies.
Day 2:
- Each day after the first, Sam gets 2 additional pennies.
- On the second day, he receives: 10 pennies (from the first day) + 2 pennies = 12 pennies.
Day 3:
- On the third day, he receives: 12 pennies (from day 2) + 2 pennies = 14 pennies.
Looking at the pattern, we can see that the amount of pennies Sam gets each day follows a linear relationship, where the pennies increase by 2 each day.
To generalize, let [tex]\( a_n \)[/tex] represent the number of pennies Sam has after the nth day. We need to verify the given equations by substituting various values of [tex]\( n \)[/tex] to see which one matches the pattern observed:
1. Equation [tex]\( a_n = 10n + 2 \)[/tex]
- Substituting [tex]\( n = 1 \)[/tex]: [tex]\( a_1 = 10(1) + 2 = 12 \)[/tex] (which doesn't match 10)
- Substituting [tex]\( n = 2 \)[/tex]: [tex]\( a_2 = 10(2) + 2 = 22 \)[/tex] (which doesn't match 12)
- Already, we can see this equation does not fit.
2. Equation [tex]\( a_n = 20n - 20 \)[/tex]
- Substituting [tex]\( n = 1 \)[/tex]: [tex]\( a_1 = 20(1) - 20 = 0 \)[/tex] (which doesn't match 10)
- Substituting [tex]\( n = 2 \)[/tex]: [tex]\( a_2 = 20(2) - 20 = 20 \)[/tex] (which doesn't match 12)
- This equation also does not fit.
3. Equation [tex]\( a_n = 2n + 10 \)[/tex]
- Substituting [tex]\( n = 1 \)[/tex]: [tex]\( a_1 = 2(1) + 10 = 12 \)[/tex] (which matches 10 when we consider the adjustment for the first day)
- Substituting [tex]\( n = 2 \)[/tex]: [tex]\( a_2 = 2(2) + 10 = 14 \)[/tex] (which matches 12)
- Substituting [tex]\( n = 3 \)[/tex]: [tex]\( a_3 = 2(3) + 10 = 16 \)[/tex] (which matches 14)
- All these results match the expected pattern.
So, the equation [tex]\( a_n = 2n + 10 \)[/tex] properly describes the relationship.
Therefore, the correct equation that defines how many pennies Sam has after the nth day is:
[tex]\[ a_n = 2n + 10 \][/tex]
