Question : If the average of 50 consecutive even numbers is 55, then find the smallest number.
Correct Answer 6
Solution & Explanation
Solution
Shortcut Trick to find the first number if the average of n consecutive numbers are givenFormula to find the first number, if the average of n consecutive numbers are given,A = x + (n –1)
or, x = A – (n – 1)
where, A = Average, x = first term, and n = number of terms
or, x = A – (n – 1)
where, A = Average, x = first term, and n = number of terms
In the question, it is given,
The average of 50 consecutive even numbers = 55
Therefore, the smallest number = ?
Here, we have A = 55
And, n = 50
Now, using formula, A = x + (n – 1)
where, A = Average, n = number of terms, and x = first term
∴ 55 = x + (50 – 1)
⇒ 55 = x + 49
⇒ x = 55 – 49 = 6
Thus, first and the smallest even number = 6 Answer
Procedure to find the numbers if the average of n consecutive even numbers are given
It is necessary to know the procedure, as it applies to solving all similar types of given problems, as shortcut tricks may not be applicable everywhere.
Given, the average of 50 consecutive even numbers = 50
Thus, the smallest and the greatest number among the given three even numbers = ?
Let first even number = a
Therefore, the second consecutive even number = a + 2
And, the third consecutive even number = a + 4
And, the fourth consecutive even number = a + 6
Now, list of these even numbers is forming a series which is
a, a+2, a+4, a+6, ..... upto 50th term
This type of series is called an Arithmetic Series
Arithmetic Series: If the difference between two consecutive terms of a series is equal, then the series is known as an Arithmetic Series
Finding the nth term of an Arithmetic Series
The nth term of an Arithmetic Series = an = a + (n – 1)d
Where, a = first term, d = common difference (difference between two consecutive terms), and n = number of terms
In the given series,
The first term (a) = a
Common difference (d)
= Second term – First term
= a + 2 – a = 2
⇒ Common difference (d) = 2
And, the number of terms (n) = 50
∴ a50 = a + (50 – 1) 2
= a + 49 × 2
⇒ a50 = a + 98
Thus, the series becomes
a, a+2, a+4, a+6, ..... a+98
Average = Sum of observations/Number of observations
Here, sum of observations (sum of numbers)
= a + a+2 + a+4 + a+6 . . . . a+98
Although, by adding all the terms we can find the sum of sum of all the given numbers, when number of terms are smaller, this can be done easily. But, when we need to add more numbers, as in this case we have to add 50 consecutive numbers, then it becomes a bit difficult. So, in such a case we will apply a formula to add all the numbers
As, we can see the list of given numbers is in the Aritmetic series, so here we will apply a formula called Sum of all terms (S)
In an Arithmetic Series, the Sum of all terms (S)
= n/2 (a + l)
where, n = number of terms, a = first term, and l = last term
Here, we have
The first term (a) = a
The last term (l) = a + 98
And, the number of terms = 50
Thus, sum of all terms (S)
= 50/2 × (a + a+98)
Thus, S (sum of all terms) = 25 (2a + 98)
Now, Average = Sum of all terms/Number of terms
⇒ 55 = 25 (2a + 98)/ 50 2
⇒ 55 = 2a + 98/2
⇒ 55 = 2 (a + 49)/ 2
⇒ 55 = a + 49
After transposing 49 to LHS
⇒ 55 – 49 = a
⇒ 6 = a
⇒ a = 6
Thus, the first term and the smallest number = 6 Answer
Similar Questions
(1) Find the average of even numbers from 6 to 946
(2) Find the average of odd numbers from 13 to 401
(3) What will be the average of the first 4201 odd numbers?
(4) What will be the average of the first 4225 odd numbers?
(5) Find the average of odd numbers from 11 to 1317
(6) What is the average of the first 426 even numbers?
(7) Find the average of odd numbers from 13 to 213
(8) Find the average of the first 4578 even numbers.
(9) Find the average of odd numbers from 9 to 1231
(10) Find the average of even numbers from 12 to 1962