Question

i. Write the next two lines of the pattern above.

ii. Write the first and the last numbers of the 10th line.

iii. Find the sum of all the numbers in the first ten lines.

Answer 1

i. First row contains one element, second row 

contains two elements, therefore j fifth row will 

contain five elements and j sixth row will contain six elements.

Consider the first row

1, 2, 4, …………..

1, 1 + 1, 2 + 2, 4 + 3, …………

Therefore

1, 1 + 1, 2 + 2, 4 + 3, 7 + 4, 11 + 5 …………..

1, 2, 4, 7, 11, 16 …………

Generally it written as

1 + 1 (1 + 2 + 3 + ……….)

Fist term in the fifth row = 1 + 1(1 + 2 + 3 + 4) = 1 + 10 = 11

Fist term in the sixth row = 1 + 1(1 + 2 + 3 + 4+ 5) = 1 + 15 = 16

Common differences in each row = 1 

Fifth row 11, 12, 13, 14, 15. 

Sixthrow 16, 17, 18, 19, 20, 21.

ii. Fist term in the tenth row = 1 + 1(1 + 2 + 3+ 4 + 5 + 6+ 7 + 8 + 9) = 1 + \(\frac{1\times9\times10}{2}=46\)

Last term in the tenth row = 46 + 1 × 9 = 46 + 9 = 55

iii. Numbers in the first ten lines 1, 2, 3, 4, 5, ………….. 55 

Sum of all the numbers in the first ten lines = \(\frac{n(n+1)}{2}=\frac{55\times56}{2}=1540\)