PRACTICE PROBLEMS WITH PATTERNS AND INDUCTION

Problem 1 :

Consider the pattern :

Find the number of matchsticks M required to make the

(a) 1^st, 2^nd and 3^rd figures

(b) 4^th and 5^th figures

(c) 10^th figure

(d) n^th figure                Solution

Problem 2 :

Consider the pattern :

2 + 4 = 6 = 2 × 3

2 + 4 + 6 = 12 = 3 × 4

2 + 4 + 6 + 8 = 20 = 4 × 5

a) Continue the pattern for 3 more cases.

b) Use a predict a formula for 2 + 4 + 6 + 8 +……+ 2n

c) Use a predict a formula for 1 + 2 + 3 + 4 +……+ n

d) what is the sum of the first 200 positive integers ?

Solution

Problem 3 :

For the following match stick pattern, find the number of matches M required to make 

(a) 4^th and 5^th figures

(b) 20^th figure

(c) n^th figure                    Solution

Problem 3 :

Consider the following pattern,

1 = 1 = 1²

1 + 3 = 4 = 2²

1 + 3 + 5 = 9 = 3²

(a) Continue the pattern for 3 more cases.

(b) Predict the value of 1 + 3 + 5 + 7 +........+ 99

(c) The 1^st odd number is 1.

The 2^nd odd number is 3.

The 3^rd odd number is 5.

What is the n^th odd number ?              Solution

Problem 5 :

For the following matchstick pattern, find the number of matches M required to make the 

(a) 8^th figure

(b) n^th figure           Solution

Problem 6 :

Consider the pattern :

S₁ = 1/(1×2)

S₂ = 1/(1×2) + 1/(2×3)

S₃ = 1/(1×2) + 1/(2×3) + 1/(3×4), …………

a) Find the values of S₁, S₂, S₃, and S₄

b) write down the value of :

(i) S₁₀  (ii) S_n             Solution

Problem 7 :

Consider the pattern :

S₁ = 1²

 S₂ = 1² + 2²

 S₂ = 1² + 2² + 3², …………

a)  Check that the formula

S_n = [n(n + 1)(2n + 1)]/6

is correct for n  =  1, 2, 3 and 4

b) Assuming the formula in a is always true, find the sum of

1² + 2² + 3² + 4² + 5² + ………… + 100² 

which is the sum of the squares of the first one hundred integers.              Solution

Problem 8 :

Consider the pattern :

N₁ = 1³ 

N₂ = 1³ + 2³

 N₃ = 1³ + 2³ + 3³, …………

a)  Verify that the formula

N_n = n²(n + 1)²/4

is correct for n = 1, 2, 3 and 4.

b) Use the above formula to find the sum of

1³ + 2³ + 3³ + 4³ + ………… + 50³

c) Find the sum : 2³ + 4³ + 6³ + 8³ + ………… + 100³

Solution

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