Question:
( 1 of 10 ) Find the average of odd numbers from 3 to 161
(A) 24
(B) 25
(C) 36
(D) 23
You selected
83
Correct Answer
82
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 3 to 161
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 3 to 161 are
3, 5, 7, . . . . 161
After observing the above list of the odd numbers from 3 to 161 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 3 to 161 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 3 to 161
The First Term (a) = 3
The Common Difference (d) = 2
And the last term (ℓ) = 161
The average of the numbers forming an Arithmetic Series
= The first term (a) + The last term (ℓ)/2
⇒ The average of numbers forming an Arithmetic Series = a + ℓ/2
Thus, the average of the odd numbers from 3 to 161
= 3 + 161/2
= 164/2 = 82
Thus, the average of the odd numbers from 3 to 161 = 82 Answer
Method (2) to find the average of the odd numbers from 3 to 161
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 3 to 161 are
3, 5, 7, . . . . 161
The odd numbers from 3 to 161 form an Arithmetic Series in which
The First Term (a) = 3
The Common Difference (d) = 2
And the last term (ℓ) = 161
The Average of the given numbers
= Sum of the given numbers/Total number of given numbers
Thus, to find the average of the given numbers, first, we need to find their sum and the total number of given numbers
Finding the number of terms
For an Arithmetic Series, the nth term
an = a + (n – 1) d
Where
a = First term
d = Common difference
n = number of terms
an = nth term
Thus, for the given series of the odd numbers from 3 to 161
161 = 3 + (n – 1) × 2
⇒ 161 = 3 + 2 n – 2
⇒ 161 = 3 – 2 + 2 n
⇒ 161 = 1 + 2 n
After transposing 1 to LHS
⇒ 161 – 1 = 2 n
⇒ 160 = 2 n
After rearranging the above expression
⇒ 2 n = 160
After transposing 2 to RHS
⇒ n = 160/2
⇒ n = 80
Thus, the number of terms of odd numbers from 3 to 161 = 80
This means 161 is the 80th term.
Finding the sum of the given odd numbers from 3 to 161
The sum of all terms (S) in an Arithmetic Series
= n/2 (a + ℓ)
Where, n = number of terms
a = First term
And, ℓ = Last term
Thus, the sum of all terms (S) of the given odd numbers from 3 to 161
= 80/2 (3 + 161)
= 80/2 × 164
= 80 × 164/2
= 13120/2 = 6560
Thus, the sum of all terms of the given odd numbers from 3 to 161 = 6560
And, the total number of terms = 80
Since, the average of the given numbers
= Sum of the given numbers/Total number of given numbers
Thus, the average of the given odd numbers from 3 to 161
= 6560/80 = 82
Thus, the average of the given odd numbers from 3 to 161 = 82 Answer
Similar Questions
(1) What will be the average of the first 4167 odd numbers?
(2) What is the average of the first 1039 even numbers?
(3) Find the average of odd numbers from 9 to 493
(4) Find the average of the first 4956 even numbers.
(5) Find the average of odd numbers from 3 to 1049
(6) Find the average of odd numbers from 13 to 1399
(7) Find the average of even numbers from 4 to 430
(8) Find the average of the first 3139 even numbers.
(9) Find the average of even numbers from 10 to 548
(10) Find the average of even numbers from 12 to 1058