## EXAMPLE PROBLEMS ON FORMULA BY INDUCTION

Example 1 :

Consider the pattern :

Find the number of matchsticks M required to make the

(a)  1st, 2nd and 3rd figures

(b)  4th and 5th figures

(c)  10th figure

(d)  nth figure

Solution :

(a)  Number of matchsticks in 1st figure  =  3

Number of matchsticks in 2nd figure  =  5

Number of matchsticks in 3rd figure  =  7

(b)  By observing the above sequence, it is very clear that every element is 2 more than the preceding. By continuing this way, we get

Number of matchsticks in 4th figure  =  (7+2)  ==> 9

Number of matchsticks in 5th figure  =  (9+2)  ==> 11

(c)  10th figure

Number of match sticks in the nth figure  =  2n+1

if n  =  10

Number of match sticks in 10th figure  =  2(10) + 1

Number of match sticks in 10th figure  =  21

(d)  nth figure

Number of match sticks in nth figure =  2n+1

Example 2 :

Consider the pattern :

2 + 4  =  6  =  2 × 3

2 + 4 + 6  =  12  =  3 × 4

2 + 4 + 6 + 8  =  20  =  4 × 5

2 terms in the first serious

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 :

By observing the serious above,

(a)  By continuing the pattern, we get

(i)  2+4+6+8+10  =  5(5+1)  ==>  30

(ii)  2+4+6+8+10+12  =  6(6+1)  ==>  42

(ii)  2+4+6+8+10+12+14  =  7(7+1)  ==>  56

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

From (a), it is clear the sum of the series

=  number of terms (number of terms + 1)

Since we have "n" terms

Sum of the series  =  n(n+1)

(c)  For example,

 Sum of series1+2+3  =  61+2+3+4  =  101+2+3+4+5  =  15 Using n(n+1)n(n+1)  =  3(3+1)  ==>  12n(n+1)  =  4(4+1)  ==>  20n(n+1)  =  5(5+1)  ==>  30Divide each case by 2

So, the required formula is n(n+1)/2.

(d)  Sum of 200 positive integers  =  200(200+1)/2

=  100(201)

=  20100

Example 3 :

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

(a)  4th and 5th figures

(b)  20th figure

(c)  nth figure

Solution :

(a)  Number of match sticks in 1st figure  =  7

Number of match sticks in 2nd figure  =  7+5  ==>  12

Number of match sticks in 3rd figure  =  12+5  ==>  17

Number of match sticks in 4th figure  =  17+5  ==>  22

Number of match sticks in 5th figure  =  22+5  ==>  27

Number of match sticks  =  multiple of 5 + 2

So, the formula is 5n+2.

(b)  Number of match sticks in 20th figure  =  5(20) + 2

=  100+2

=  102

(c)  The required formula is 5n+2.

Example 4 :

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 1st odd number is 1.

The 2nd odd number is 3.

The 3rd odd number is 5.

What is the nth odd number ?

Solution :

(a)  (i)  1+3+5+7  =  16  =  42

(ii)  1+3+5+7+9  =  25  =  52

(iii)  1+3+5+7+9+11  =  36  =  62

"n" number of terms in each cases.

(b)  Number of terms from 1 to 100 is 100.

Number of odd terms from 1 to 99 is 50.

=  502

=  2500

(c)  Each terms are odd numbers.

Formula  =  2n-1

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