Question:
Find the average of odd numbers from 13 to 1189
Correct Answer
601
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 13 to 1189
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 13 to 1189 are
13, 15, 17, . . . . 1189
After observing the above list of the odd numbers from 13 to 1189 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 13 to 1189 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 13 to 1189
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 1189
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 13 to 1189
= 13 + 1189/2
= 1202/2 = 601
Thus, the average of the odd numbers from 13 to 1189 = 601 Answer
Method (2) to find the average of the odd numbers from 13 to 1189
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 13 to 1189 are
13, 15, 17, . . . . 1189
The odd numbers from 13 to 1189 form an Arithmetic Series in which
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 1189
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 13 to 1189
1189 = 13 + (n – 1) × 2
⇒ 1189 = 13 + 2 n – 2
⇒ 1189 = 13 – 2 + 2 n
⇒ 1189 = 11 + 2 n
After transposing 11 to LHS
⇒ 1189 – 11 = 2 n
⇒ 1178 = 2 n
After rearranging the above expression
⇒ 2 n = 1178
After transposing 2 to RHS
⇒ n = 1178/2
⇒ n = 589
Thus, the number of terms of odd numbers from 13 to 1189 = 589
This means 1189 is the 589th term.
Finding the sum of the given odd numbers from 13 to 1189
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 13 to 1189
= 589/2 (13 + 1189)
= 589/2 × 1202
= 589 × 1202/2
= 707978/2 = 353989
Thus, the sum of all terms of the given odd numbers from 13 to 1189 = 353989
And, the total number of terms = 589
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 13 to 1189
= 353989/589 = 601
Thus, the average of the given odd numbers from 13 to 1189 = 601 Answer
Similar Questions
(1) What is the average of the first 897 even numbers?
(2) What will be the average of the first 4888 odd numbers?
(3) Find the average of the first 3160 even numbers.
(4) Find the average of the first 3353 even numbers.
(5) Find the average of even numbers from 8 to 234
(6) Find the average of odd numbers from 3 to 61
(7) Find the average of the first 726 odd numbers.
(8) Find the average of the first 2056 even numbers.
(9) What is the average of the first 1584 even numbers?
(10) Find the average of even numbers from 6 to 1598