Question

Express 49 as the sum of 7 odd numbers.

Answer 1

Hint: In this question, we first need to look at some of the basic definitions of the number system. Then from the definition of odd numbers we know that the sum of first n odd numbers is given by $n^2$. Now, by using this formula we need to write 49 as the sum of 7 odd numbers.

Let us look at some of the basic definitions of the number system.

NUMBER: A number tells us how many times a unit is contained in a given quantity.

NUMERAL: A group of figures or digits, representing a number is called a numeral.

DECIMAL NUMBER SYSTEM: Numeric values are represented by using digits from 0 to 9.

Again numbers in the decimal system are classified as real numbers.

Real numbers are classified into rational numbers and irrational numbers.

Rational numbers are again classified into integer and non-integer rational numbers.

Again integers are classified into:

Positive integer
Negative integer
Whole number
Natural number

The decimal number system is the standard system for denoting integer and non-integer numbers.

Complete step-by-step answer:
NATURAL NUMBERS: Numbers starting from 1, having no fraction part, which we use in counting the objects, denoted by N.

\[N=\left\{ 1,2,3...... \right\}\]

WHOLE NUMBERS: The system of Natural numbers along with number 0, is called whole number and is denoted by W.

\[W=\left\{ 0,1,2,3...... \right\}\]

Different Types of Natural Number:

EVEN NUMBER: A number, which is multiple of 2 is called an even number.

ODD NUMBER: A number, which is not multiple of 2 is called an odd number.

Now, let us consider the sum of the first two odd numbers.

\[\Rightarrow 1+3=4={{2}^{2}}\]

Let us now consider the sum of the first 3 odd numbers.

\[\Rightarrow 1+3+5=9={{3}^{2}}\]

Now, consider the sum of the first 4 odd numbers.

\[\Rightarrow 1+3+5+7=16={{4}^{2}}\]

Thus, from the above considerations we can generalise and get the sum of first n odd numbers.

Now, we know that sum of first n odd numbers is given by:

\[{{n}^{2}}\]

49 can also be written as a square of 7.

\[\Rightarrow 49={{7}^{2}}\]

Thus, we can say that the sum of the first 7 odd numbers is 49.

\[\therefore 49=1+3+5+7+9+11+13\]

Note: It is important to note that 49 can be expressed as a square of a number and that number is 7 because we were asked to write it as a sum of 7 odd numbers. This simplifies as 49 can be expressed as the sum of first n odd numbers which we can directly express.
There are many possible ways to express 49 as a sum of other numbers which can be either even or odd. But, we are supposed to express here a sum of 7 odd numbers which can be done only in one way that is the sum of the first 7 odd numbers.