NCERT Class 8 Mathematics Ganita Prakash Part 1 Chapter 05 Number Play PDF Download

Chapter 5: Number Play

5.1 Is This a Multiple Of?

Sum of Consecutive Numbers

Anshu is exploring sums of consecutive numbers. He has written the following -

\( 7 = 3 + 4 \)
\( 10 = 1 + 2 + 3 + 4 \)
\( 12 = 3 + 4 + 5 \)
\( 15 = 7 + 8 \)
\( = 4 + 5 + 6 \)
\( = 1 + 2 + 3 + 4 + 5 \)

Now, he is wondering -

• "Can I write every natural number as a sum of consecutive numbers?"
• "Which numbers can I write as the sum of consecutive numbers in more than one way?"
• "Ohh, I know all odd numbers can be written as a sum of two consecutive numbers. Can we write all even numbers as a sum of consecutive numbers?"
• "Can I write 0 as a sum of consecutive numbers? Maybe I should use negative numbers."

Explore these questions and any others that may occur to you. Discuss them with the class.

Take any 4 consecutive numbers. For example, 3, 4, 5, and 6. Place '+' and '-' signs in between the numbers. How many different possibilities exist? Write all of them.

\( 3 + 4 - 5 + 6 \)
\( 3 - 4 - 5 - 6 \)

Eight such expressions are possible. You can use the diagram below to systematically list all the possibilities.

[Figure: Tree diagram showing 8 different sign combinations for 3, 4, 5, and 6, See in your textbook]

Evaluate each expression and write the result next to it. Do you notice anything interesting?

Now, take four other consecutive numbers. Place the '+' and '-' signs as you have done before. Find out the results of each expression. What do you observe?

Repeat this for one more set of 4 consecutive numbers. Share your findings.

\( 3 + 4 - 5 + 6 = 8 \)
\( 3 - 4 - 5 - 6 = -12 \)

\( 5 + 6 - 7 + 8 = 12 \)
\( 5 - 6 - 7 - 8 = -16 \)

\( \_\_ + \_\_ - \_\_ + \_\_ = \_\_ \)
\( \_\_ - \_\_ - \_\_ - \_\_ = \_\_ \)

Some sums appear always no matter which 4 consecutive numbers are chosen. Isn't that interesting?

Do these patterns occur no matter which 4 consecutive numbers are chosen? Is there a way to find out through reasoning?

Hint: Use algebra and describe the 8 expressions in a general form.

You might have noticed that the results of all expressions are even numbers. Even numbers have a factor of 2. Negative numbers having a factor 2 are also even numbers, for example, \( -2 \), \( -4 \), \( -6 \), and so on. Check if anyone in your class got an odd number.

When 4 consecutive numbers are chosen, no matter how the '+' and '-' signs are placed between them, the resulting expressions always have even parity.

Teacher's Note

When switching a sign from \( + \) to \( - \) (or vice versa) in an expression like \( a + b - c - d \), the change in value is always even. This is why all 8 expressions must have the same parity. If you get different parities in your calculations, check for arithmetic errors.

Now take any 4 numbers, place '+' and '-' signs in the eight different ways, and evaluate the resulting expression. What do you observe about their parities?

Repeat this with other sets of 4 numbers.

Is there a way to explain why this happens?

Hint: Think of the rules for parity of the sum or difference of two numbers.

Explanation 1: Let us consider any of the 8 expressions formed by four numbers \( a \), \( b \), \( c \), and \( d \). When one of its signs is switched, its value always increases or decreases by an even number! Let us see why.

Consider one of the expressions: \( a + b - c - d \).

Replacing \( +b \) by \( -b \), we get

\[ a - b - c - d. \]

By how much has the number changed? It has changed by

\[ (a + b - c - d) - (a - b - c - d) \] \[ = a + b - c - d - a + b + c + d \text{ (notice how the signs changed when we opened the second set of brackets)} \] \[ = 2b \text{ (this is an even number)}. \]

If the difference between two numbers is even, can they have different parities? No! So either both are even or both are odd.

Now, let us see what happens when a negative sign is switched to a positive sign.

Replace any negative sign in the expression \( a + b - c - d \) with a positive sign and find the difference between the two numbers.

What do you conclude from this observation?

Starting from any expression, we can get 7 expressions by switching one or more '+' and '-' signs. Thus, all the expressions have the same parity!

Explanation 2: We know that

\[ \text{odd} \pm \text{odd} = \text{even} \] \[ \text{even} \pm \text{even} = \text{even} \] \[ \text{odd} \pm \text{even} = \text{odd}. \]

We have seen that the parity of \( a + b \) and \( a - b \) is the same, regardless of the parities of \( a \) and \( b \).

In short, \( a \pm b \) have the same parity. By the same argument, \( a \pm b + c \) and \( a \pm b - c \) have the same parity. Extending this further, we can say that all the expressions \( a \pm b \pm c \pm d \) have the same parity.

Teacher's Note

Parity is about whether a number is odd or even, and this property is preserved when you add or subtract. Remember that \( \text{odd} + \text{odd} = \text{even} \) and \( \text{even} + \text{even} = \text{even} \), but \( \text{odd} + \text{even} = \text{odd} \). This is the key rule that makes all eight expressions have the same parity no matter what signs you use.

Key Points

• When four numbers are combined with addition and subtraction signs in any of the eight ways, the result always has the same parity (all even or all odd).
• Changing a plus sign to a minus sign (or vice versa) changes the value by an even number, which preserves parity across all expressions.
• The parity rules are: odd plus or minus odd gives even, even plus or minus even gives even, and odd plus or minus even gives odd.
• Since \( a + b \) and \( a - b \) always have the same parity, this property extends to expressions with more terms like \( a \pm b \pm c \pm d \).

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