Since no specific problem was selected from the worksheet, I will solve
Problem 22 as it is a distinct word problem involving arithmetic sequences, which is a common type of question on these reviews.
Problem 22: "A display of soup cans is arranged with 1 can on top and each row having an additional can. How many cans are in a display of 20 rows?"
Step-by-Step Deduction
1.
Identify the Pattern:
* The problem states there is
1 can in the top row.
* Each subsequent row has one additional can. This means:
* Row 1 has 1 can.
* Row 2 has 2 cans.
* Row 3 has 3 cans.
* ...and so on.
* We need to find the total number of cans for
20 rows. This means we need to add the numbers from 1 to 20:
$$1 + 2 + 3 + ... + 20$$
2.
Choose the Method:
* This is an
arithmetic series (a sequence of numbers with a constant difference).
* We can use the summation formula for an arithmetic series:
$$S_n = \frac{n}{2}(a_1 + a_n)$$
Where:
* $n$ is the number of terms (rows).
* $a_1$ is the first term (cans in the first row).
* $a_n$ is the last term (cans in the last row).
3.
Identify the Variables:
* $n = 20$ (since there are 20 rows).
* $a_1 = 1$ (the top row has 1 can).
* $a_{20} = 20$ (since the row number equals the number of cans, the 20th row has 20 cans).
4.
Perform the Calculation:
* Substitute the values into the formula:
$$S_{20} = \frac{20}{2}(1 + 20)$$
* Simplify the fraction and the parentheses:
$$S_{20} = 10(21)$$
* Multiply:
$$10 \times 21 = 210$$
5.
Verification:
* Alternatively, using the formula $S_n = \frac{n(n+1)}{2}$:
$$S_{20} = \frac{20(20+1)}{2} = \frac{20 \times 21}{2} = \frac{420}{2} = 210$$
* Both methods yield the same result.
Final Answer
There are
210 cans in the display.
Parent Tip: Review the logic above to help your child master the concept of algebra 2 review worksheet.
