Sequences, Series, and Limits Quiz

Grade 11 · Math · Quiz

Test your mastery of arithmetic and geometric sequences, series, and limit concepts.

Questions

1. The first term of a geometric sequence is 4 and the common ratio is 3. What is the 5th term?

   1. 108
   2. 324
   3. 243
   4. 972
   Show answer

   Answer: 324

   a_5 = 4 * 3^4 = 4 * 81 = 324.

2. What is the sum of the first 6 terms of the arithmetic series with a_1 = 2 and d = 4?

   1. 62
   2. 72
   3. 52
   4. 82
   Show answer

   Answer: 72

   a_6 = 2 + 5*4 = 22; S_6 = 6/2 * (2 + 22) = 3 * 24 = 72.

3. Which of the following sequences is convergent?

   1. a_n = 2^n
   2. a_n = (-1)^n
   3. a_n = n^2
   4. a_n = 1/n^2
   Show answer

   Answer: a_n = 1/n^2

   1/n^2 → 0 as n → ∞, so it converges. The others grow without bound or oscillate.

4. What is lim(n→∞) (3n^2 + 2n) / (n^2 - 5)?

   1. 0
   2. 2
   3. 3
   4. Undefined
   Show answer

   Answer: 3

   Divide numerator and denominator by n^2: (3 + 2/n)/(1 - 5/n^2) → 3/1 = 3.

5. For the geometric series 8 - 4 + 2 - 1 + ..., what is the sum to infinity?

   1. 16/3
   2. 4
   3. 16
   4. 16/3 only if |r|<1 which it is not
   Show answer

   Answer: 16/3

   a_1 = 8, r = -1/2, |r| < 1. S = 8/(1-(-1/2)) = 8/(3/2) = 16/3.

6. Which formula gives the sum of the first n natural numbers?

   1. n(n+1)
   2. n(n+1)/2
   3. n^2
   4. n(n-1)/2
   Show answer

   Answer: n(n+1)/2

   The sum of the first n natural numbers is n(n+1)/2, derived from the arithmetic series formula.

7. The series 1 + 1/2 + 1/4 + 1/8 + ... has a finite sum.

   Show answer

   Answer: True

   This is an infinite geometric series with a_1=1 and r=1/2. Since |r|<1, S=1/(1-1/2)=2.

8. If a geometric sequence has a_1 = 5 and r = 2, the sum of the first 4 terms is ___.

   Show answer

   Answer: 75

   S_4 = 5(1-2^4)/(1-2) = 5(1-16)/(-1) = 5*(-15)/(-1) = 75.

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