IMO Patterns and Sequences 3

Question 1

What is the missing term: 100, 81, 64, ___, 36?

Accepted Answer

49

Answer

These are perfect squares in decreasing order: 10², 9², 8², 7², 6²; the missing term is 7² = 49.

Question 2

The rule for a sequence is: multiply the previous term by 2, then subtract 1. If the first term is 4, what is the 4th term?

Accepted Answer

25

Answer

T1=4, T2=4×2−1=7, T3=7×2−1=13, T4=13×2−1=25.

Question 3

How many terms are in the arithmetic sequence: 7, 11, 15, ..., 55?

Accepted Answer

13

Answer

n = (last − first)/d + 1 = (55 − 7)/4 + 1 = 48/4 + 1 = 13.

Question 4

A pattern shows triangular numbers: 1, 3, 6, 10, 15, ___. What is the next triangular number?

Accepted Answer

21

Answer

Triangular numbers are 1, 3, 6, 10, 15, 21 — each adding the next natural number: +2, +3, +4, +5, +6.

Question 5

The sum of an arithmetic series is given by S = n/2 (a + l). Find the sum of 1+2+3+...+100.

Accepted Answer

5050

Answer

S = 100/2 × (1 + 100) = 50 × 101 = 5050.

IMO - International Mathematics Olympiad Practice Test