## NCERT Class 10 Maths Lab Manual – Arithmetic Progression II

**Objective**

To verify that the sum of first n natural numbers is \(\frac { n(n+1) }{ 2 } \) by graphical method.

**The product of two polynomials say A and B represents a rectangle of sides A and B. Thus n(n+1) represents a rectangle of sides n and (n + 1).**

**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 in fig. (i).
- Count the coloured squares and total number of squares in rectangle PQRS.

**Observation
**We observe, number of shaded squares = \(\frac { 1 }{ 2 } \) 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 = \(\frac { 1 }{ 2 } \) n(n + 1)

**Mathematically**

Area of rectangle PQRS = 10 x 11

Area of shaded region = \(\frac { 1 }{ 2 } \) 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= \(\frac { 1 }{ 2 } \) x 10 x 11 = 55

Verified that 1 + 2 + 3 + … + 10 = \(\frac { 1 }{ 2 } \) x 10 (10 + 1) by graphical method.

**Result**

It is verified graphically that 1 + 2 + 3 + … + n = \(\frac { 1 }{ 2 } \)n(n+ 1) or sum of first n natural numbers = \(\frac { 1 }{ 2 } \) 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**

1. Find the sum of first 100 natural numbers.

2. Find the sum of first 1000 natural numbers.

3. Evaluate 10 + 11 + 12 + … + 25.

You can also download **NCERT Solutions Class 10 Maths** to help you to revise complete syllabus and score more marks in your examinations.

**Viva Voce**

**Question 1:**

Are all natural numbers whole numbers ?

**Answer:**

Yes

**Question 2:**

Are all whole numbers natural numbers ?

**Answer:**

Except zero, all whole numbers are natural numbers.

**Question 3:**

Write down an AP having the sum of first 7 terms as zero.

**Answer:**

-3, -2, -1, 0, 1, 2, 3.

**Question 4:**

What does represent, where S„ represents the sum of n terms of anAP?

**Answer:**

The n th term of an AP.

**Question 5:**

What is the formula for the sum of n terms of an AP ?

**Answer:**

S_{n}=\(\frac { n }{ 2 } [2a+(n-1)d]\)

**Question 6:**

What is the formula for the sum of n terms of an AP whose common difference is not given ? [First term (a) and last term (i) known]

**Answer:**

S_{n}=\(\frac { n }{ 2 } [a+l]\), where l represents the last term.

**Question 7:**

If S_{n }= 3n^{2}+2n, find the first term.

**Answer:**

5

**Question 8:**

What is the arithmetic mean of 4 and 8 ?

**Answer:**

6

**Question 9:**

What is the sum of first 10 natural numbers ?

**Answer:**

55

**Question 10:**

Find the common difference of an arithmetic progression of first 20 natural numbers.

**Answer:**

1

**Multiple Choice Questions**

**Question 1:**

Sum of first n terms of an AP is

(a) \(\frac { n }{ 2 } [2a+(n-1)d]\)

(b) \(\frac { n }{ 2 }2n [a+(n-1)d]\)

(c) \(\frac { n }{ 2 } [2a-(n-1)d]\)

(d) \(\frac { n }{ 2 } [2a-(n+1)d]\)

**Question 2:**

Sum of first n positive integers is

(a) \(\frac { n(n-1) }{ 2 } \)

(b) \(\frac {2 n(n+1) }{ 2 } \)

(c) \(\frac { n(n+1) }{ 2 } \)

(d) none of these

**Question 3:**

The sum of 0.70 + 0.71 + 0.72 + ….. + to 50 terms is

(a) 4.725

(b) 47.25

(c) 472.5

(d) none of these

**Question 4:**

If a_{n }= 3 + 4n is n th termof an AP, then S15 is

(a) 525

(b) 325

(c) 425

(d) none of these

**Question 5:**

Sum of all odd numbers between 0 and 50 is

(a) 623

(b) 627

(c) 624

(d) 625

**Question 6:**

Sum of -37, -33, -29, … to 12 terms is

(a) -180

(b) 180

(c) 108

(d) -108

**Question 7:**

In an AP, given that a12 = 37 and d= 3. Find S12.

(a) 246

(b) 642

(c) 264

(d) 624

**Question 8:**

In an AP, if a = 8, a_{n } = 62 and Sn = 210, then n is

(a) 4

(b) 6

(c) 5

(d) 7

**Question 9:**.

Sum of first 40 positive integers divisible by 6 is

(a) 4092

(b) 4029

(c) 4920

(d) 4290

**Question 10:**

Sum of first 15 multiples of 8 is

(a) 690

(b) 609

(r) 906

(d) 960

**Answers**

1. (a)

2. (c)

3. (b)

4. (a)

5 (d)

6. (a)

7. (a)

8. (b)

9. (c)

10. (d)

Math LabsScience LabsScience Practical SkillsMath Labs with Activity