Lab Manual: Arithmetic Progression II | Lab Manuals for Class 10 PDF Download

Objective

Prerequisite Knowledge

• Concept of natural numbers.
• Area of squares and rectangles.

Materials Required
Graph papers, white chart paper, coloured pens, geometry box.

Procedure
Let us consider the sum of first n natural numbers 1 + 2 + 3 + 4 + + n (say n = 10).

• Take a graph paper and paste it on a white chart paper.
• Mark the rectangles 1, 2, 3 n, (n + 1) along the vertical line and 1,2, 3,…. n along the horLontal line.
•  Colour the rectangular strips of length 1 cm, 2 cm, 3 cm n cm each of width 1 cm.
•  Complete the rectangle with sides n and n+1. Name this rectangle as PQRS. Mark dot in each square as shown.
•  Count the coloured squares and total number of squares in rectangle PQRS.

Observation
We observe, number of shaded squares =(1/2) x total no. of squares
No. of shaded squares = 1+ 2 + 3 + … + n
Total squares = Area of rectangle = n (n + 1)
Therefore 1 + 2 + 3 + … + n = (1/2)n(n + 1)

Mathematically
Area of rectangle PQRS = 10 x 11
Area of shaded region = (1/2) x 10 x 11 = 55 ……………….(i)
Also, area of shaded region = (1 x 1) + (2 x 1) + (3 x 1) +… + (10 x 1)
= 1+2 + 3 + … +10 = 55 …………………….(ii)
From (i) and (ii),
1+2 + 3 + … + 10=(1/2)x 10 x 11 = 55
Verified that 1 + 2 + 3 + … + 10 = (1/2)x 10 (10 + 1) by graphical method.

Result
It is verified graphically that 1 + 2 + 3 + … + n = (1/2)n(n+ 1) or sum of first n natural numbers =(1/2) n(n + 1).

Learning Outcome
Students will develop a geometrical intuition of the formula for the sum of natural numbers starting from one.

Activity Time

• Find the sum of first 100 natural numbers.
• Find the sum of first 1000 natural numbers.
• Evaluate 10 + 11 + 12 + … + 25.