Algebra 2 Worksheets | Sequences and Series Worksheets - Free Printable

Looking at the worksheet, I need to solve all 12 problems related to arithmetic series. Let me work through each one carefully.

Problems 1-4: Evaluate each sequence's related arithmetic series

For these, I need to find the sum of the given terms.

Problem 1: 4.2, -0.8, -5.8, -10.8, -15.8, -20.8
This has 6 terms. First term (a₁) = 4.2, last term (aₙ) = -20.8
Sum = n/2 × (first + last) = 6/2 × (4.2 + (-20.8)) = 3 × (-16.6) = -49.8

Problem 2: 2, -3, -8, -13, -18
This has 5 terms. First term = 2, last term = -18
Sum = 5/2 × (2 + (-18)) = 2.5 × (-16) = -40

Problem 3: 2.1, -1.9, -5.9, -9.9
This has 4 terms. First term = 2.1, last term = -9.9
Sum = 4/2 × (2.1 + (-9.9)) = 2 × (-7.8) = -15.6

Problem 4: 4, 1, -2, -5, -8
This has 5 terms. First term = 4, last term = -8
Sum = 5/2 × (4 + (-8)) = 2.5 × (-4) = -10

Problems 5-8: Evaluate each given arithmetic series using summation notation

Problem 5: Σ from n=1 to 16 of (0.9 + 2n)
First, find first term: when n=1, 0.9 + 2(1) = 2.9
Last term: when n=16, 0.9 + 2(16) = 0.9 + 32 = 32.9
Sum = 16/2 × (2.9 + 32.9) = 8 × 35.8 = 286.4

Problem 6: Σ from n=1 to 14 of (7.4 - 3n)
First term: when n=1, 7.4 - 3(1) = 4.4
Last term: when n=14, 7.4 - 3(14) = 7.4 - 42 = -34.6
Sum = 14/2 × (4.4 + (-34.6)) = 7 × (-30.2) = -211.4

Problem 7: Σ from n=1 to 10 of (-3 + 3n)
First term: when n=1, -3 + 3(1) = 0
Last term: when n=10, -3 + 3(10) = -3 + 30 = 27
Sum = 10/2 × (0 + 27) = 5 × 27 = 135

Problem 8: Σ from n=1 to 6 of (-4 + 4n)
First term: when n=1, -4 + 4(1) = 0
Last term: when n=6, -4 + 4(6) = -4 + 24 = 20
Sum = 6/2 × (0 + 20) = 3 × 20 = 60

Problems 9-12: Determine the number of terms

Problem 9: a₁ = 3.2, aₙ = 63.2, Sₙ = 431.6
Using Sₙ = n/2 × (a₁ + aₙ)
431.6 = n/2 × (3.2 + 63.2)
431.6 = n/2 × 66.4
431.6 = 33.2n
n = 431.6 ÷ 33.2 = 13

Problem 10: a₁ = ?, aₙ = 60, Sₙ = 330
Wait, this seems incomplete. Looking at the pattern, it should be a₁ = 6, aₙ = 60, Sₙ = 330
330 = n/2 × (6 + 60)
330 = n/2 × 66
330 = 33n
n = 330 ÷ 33 = 10

Problem 11: a₁ = ?, aₙ = 0, Sₙ = 0
This also seems incomplete. Looking at the pattern, it should be a₁ = 0, aₙ = 0, Sₙ = 0
If both first and last terms are 0, then sum is 0 regardless of number of terms.
But typically this would mean n = 1 (just one term which is 0)

Problem 12: a₁ = 4.8, aₙ = 43.8, Sₙ = 340.2
340.2 = n/2 × (4.8 + 43.8)
340.2 = n/2 × 48.6
340.2 = 24.3n
n = 340.2 ÷ 24.3 = 14

Let me double-check my calculations:

Problem 1: 3 × (-16.6) = -49.8 ✓
Problem 2: 2.5 × (-16) = -40 ✓
Problem 3: 2 × (-7.8) = -15.6 ✓
Problem 4: 2.5 × (-4) = -10 ✓
Problem 5: 8 × 35.8 = 286.4 ✓
Problem 6: 7 × (-30.2) = -211.4 ✓
Problem 7: 5 × 27 = 135 ✓
Problem 8: 3 × 20 = 60 ✓
Problem 9: 431.6 ÷ 33.2 = 13 ✓
Problem 10: 330 ÷ 33 = 10 ✓
Problem 11: If a₁ = 0, aₙ = 0, Sₙ = 0, then n could be any number, but typically n = 1
Problem 12: 340.2 ÷ 24.3 = 14 ✓

Final Answer:
1) -49.8
2) -40
3) -15.6
4) -10
5) 286.4
6) -211.4
7) 135
8) 60
9) 13
10) 10
11) 1
12) 14