Sequences & series worksheet with answers

Free printable Sequences & series worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy
kak​123456789101112

The arithmetic sequence 15, 16.5, 18, 19.5, … ends with the term 45. How many terms n does it have?

a₁ = 15d = 1.5aₙ = 45
n =
2.Easy
kak​13579111315ε = 0.2

The geometric sequence aₙ = 8·0.75ⁿ⁻¹ decreases. Find the first index n for which aₙ < 0.2.

a₁ = 8r = 0.75ε = 0.2
n =
3.Medium
k|aₖ₊₁/aₖ|123456789101112

Apply the ratio test to Σk=1∞ kᵏ/(5ᵏ·k!). Find L = lim |ak+1/ak|.

ak = kᵏ/(5ᵏ·k!)
L =
4.Medium
k|aₖ₊₁/aₖ|123456789101112

Apply the ratio test to Σk=1∞ k·5ᵏ/2ᵏ. Find L = lim |ak+1/ak|.

ak = k·5ᵏ/2ᵏ
L =
5.Hard

f(x) = eˣ is approximated at x = 2 by its Maclaurin polynomial of degree n. With the Lagrange bound M·|x|ⁿ⁺¹/(n + 1)! and M = 8, find the smallest degree n that guarantees an error below 1 × 10⁻⁵.

f(x) = eˣx = 2M = 8ε = 1 × 10⁻⁵
n =
6.Hard
xyfP₃0.5

Use the Maclaurin polynomial of degree 3 of f(x) = 1/(1 − x) to approximate f(0.5). Give P₃(0.5).

f(x) = 1/(1 − x)n = 3x = 0.5
P₃(0.5) =

Answer key

  1. 1. n = 21
  2. 2. n = 14
  3. 3. L = 0.5437
  4. 4. L = 2.5
  5. 5. n = 13
  6. 6. P₃(0.5) = 1.875

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