Question

The average of $7$ consecutive numbers is $20$. What is the largest number?

Answer 1

Hint: The consecutive numbers are the numbers that are a step of the previous number. The consecutive numbers are in ascending order. The average is nothing but the sum of the numbers divided by the total number of numbers that are added upon the numerator. By arranging the numbers in ascending order, the last number is greater.
 Here we will take ‘x’ as the first number and will add 1 to get the next number. By this we will get all the 7 numbers. Then we will find the average which will be equal to 20. We will get a linear equation of one variable. By solving this we will get the value of ‘x’. Then we will find the last number as this will be the largest number.
Average $A = \dfrac{{{x_1} + {x_2} + ........ + {x_n}}}{n}$,
n is the total number of values.

Complete step by step solution:
Given that the total number of values is $7$.
Let us consider the first number to be $x$.
The second number of the consecutive series is $x + 1$.
The third number of the consecutive series is $x + 2$.
The fourth number of the consecutive series is $x + 3$.
The fifth number of the consecutive series is $x + 4$.
The sixth number of the consecutive series is $x + 5$.
The seventh number of the consecutive series is $x + 6$.
The average of the consecutive series is $20$.
Let us consider \[{x_1} = x\]
${x_2} = x + 1$
${x_3} = x + 2$
${x_4} = x + 3$
${x_5} = x + 4$
${x_6} = x + 5$
${x_7} = x + 6$
The total number of values is $7$.
Let us consider $n$as the total number of values
$n = 7$
By substituting values \[{x_1} = x\],${x_2} = x + 1$,${x_3} = x + 2$,${x_4} = x + 3$,${x_5} = x + 4$,${x_6} = x + 5$, ${x_7} = x + 6$and $n = 7$in the formula $A = \dfrac{{{x_1} + {x_2} + ........ + {x_n}}}{n}$.
$A = \dfrac{{x + x + 1 + x + 2 + x + 3 + x + 4 + x + 5 + x + 6}}{7}$
By adding the like terms in the numerator, we get
$A = \dfrac{{7x + 21}}{7}$
By taking the common values in the numerator, we get
$A = \dfrac{{7(x + 3)}}{7}$
By canceling the numerator and denominator, we get
$A = x + 3$
By the given that the average of $7$ consecutive number is $20$,
$A = 20$
By substituting $A = 20$in $A = x + 3$, we get
$20 = x + 3$
By subtracting the value in the above equation,
$x = 17$.
The numbers are $17,18,19,20,21,22$and $23$.
The largest number is $23$.

Note: The average is nothing but the sum of the number divided by the total number of numbers that are added upon the numerator. It should not be multiplied by the total number of numbers that are added to the numerator. It must be added on the numerator nor subtracted nor multiplied.