Question

Natural numbers are divided into groups as follows {1}, {2, 3}, {4, 5, 6}, {7, 8, 9, 10}…... Find i) Last number of 200th group ii) First number of 300th group iii) Sum of numbers in 100th group

Answer 1

Given:

The divided groups - {1}, {2, 3}, {4, 5, 6}, {7, 8, 9, 10}

To Find:

i) Last number of 200th group

ii) First number of 300th group

iii) Sum of numbers in 100th group

Solution:

Let S =1+2+4+7+...+Tn  

or S = 1+2+4+...+Tn−1+Tn  

On subtracting, we will get  

O = 1+[1+2+3+...(n-1)]-Tn  

= Tn = 1+2+3+.....+(n-1)+1  

= n (n-1)/2 + 1

1. Last number of 200th term -

= 200 × 199/2 + 200

= 20100

2. First number of 300th term

= 300 × 299/2 + 1

= 44851

3. First number of the 100th term will be -  

= 100 × 99/2 + 1  

= 4951

Similarly the sum of numbers of the 100th term -  

4951 + 4952 + ...upto 100th

Term -  

= 100/2 ( 2 × 4951 + ( 100 - 1) × 1 )

= 50 ( 9902 + 99)

= 50 × 9902

= 495100

Answer: The sum of the numbers in 100th group is 495100.