Quiz & Worksheet - Summation Notation & Mathematical Series ... - Free Printable

Let’s solve each problem one at a time, step by step.

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Problem 1: Evaluate the sum ∑ from n=2 to 4 of (3n - 4)

This means we plug in n = 2, then n = 3, then n = 4 into the expression “3n - 4”, and add up all the results.

Step 1: When n = 2
→ 3(2) - 4 = 6 - 4 = 2

Step 2: When n = 3
→ 3(3) - 4 = 9 - 4 = 5

Step 3: When n = 4
→ 3(4) - 4 = 12 - 4 = 8

Now add them: 2 + 5 + 8 = 15

✔ So the answer is 15

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Problem 2: Express the series 3 + 5 + 9 + 17 + 33 using sigma notation

We need to find a pattern that gives us these numbers when we plug in n = 1, 2, 3, 4, 5.

Let’s test the first option:
∑ from n=1 to 5 of (2ⁿ + 1)

Check each term:

- n=1 → 2¹ + 1 = 2 + 1 = 3 ✔
- n=2 → 2² + 1 = 4 + 1 = 5 ✔
- n=3 → 2³ + 1 = 8 + 1 = 9 ✔
- n=4 → 2⁴ + 1 = 16 + 1 = 17 ✔
- n=5 → 2⁵ + 1 = 32 + 1 = 33 ✔

Perfect! That matches exactly.

The other options either go to infinity (which doesn’t match since we only have 5 terms), or use wrong formulas like “3 + 2n” which would give 5, 7, 9... not matching.

✔ So the correct answer is: ∑ from n=1 to 5 of (2ⁿ + 1)

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Problem 3: Find the rule for the series: 2 + 6 + 18 + 54 + ...

Look at how each term changes:

- Start with 2
- Next: 2 × 3 = 6
- Then: 6 × 3 = 18
- Then: 18 × 3 = 54

So it’s multiplying by 3 each time — this is a geometric sequence.

In a geometric sequence, the formula is:
aₙ = a₁ × r^(n−1)
where a₁ is the first term, r is the common ratio.

Here:
a₁ = 2
r = 3

So:
aₙ = 2 × 3^(n−1)

Let’s check:

- n=1 → 2 × 3⁰ = 2 × 1 = 2 ✔
- n=2 → 2 × 3¹ = 2 × 3 = 6 ✔
- n=3 → 2 × 3² = 2 × 9 = 18 ✔
- n=4 → 2 × 3³ = 2 × 27 = 54 ✔

Perfect match.

Now look at the choices:

- aₙ = 3(2)^(n−1) → starts with 3, too big ✘
- aₙ = 2(3)^(n−1) → YES ✔
- aₙ = n² + 2 → n=1 → 1+2=3 ✘
- aₙ = n + 4 → n=1 → 5 ✘

✔ Correct answer: aₙ = 2(3)^(n−1)

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Final Answer:
1. 15
2. ∑ from n=1 to 5 of (2ⁿ + 1)
3. aₙ = 2(3)^(n−1)