Question:
( 1 of 10 ) Find the average of odd numbers from 15 to 1579
(A) 24
(B) 25
(C) 36
(D) 23
You selected
798
Correct Answer
797
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 15 to 1579
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 15 to 1579 are
15, 17, 19, . . . . 1579
After observing the above list of the odd numbers from 15 to 1579 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 1579 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 15 to 1579
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 1579
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 15 to 1579
= 15 + 1579/2
= 1594/2 = 797
Thus, the average of the odd numbers from 15 to 1579 = 797 Answer
Method (2) to find the average of the odd numbers from 15 to 1579
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 15 to 1579 are
15, 17, 19, . . . . 1579
The odd numbers from 15 to 1579 form an Arithmetic Series in which
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 1579
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 15 to 1579
1579 = 15 + (n – 1) × 2
⇒ 1579 = 15 + 2 n – 2
⇒ 1579 = 15 – 2 + 2 n
⇒ 1579 = 13 + 2 n
After transposing 13 to LHS
⇒ 1579 – 13 = 2 n
⇒ 1566 = 2 n
After rearranging the above expression
⇒ 2 n = 1566
After transposing 2 to RHS
⇒ n = 1566/2
⇒ n = 783
Thus, the number of terms of odd numbers from 15 to 1579 = 783
This means 1579 is the 783th term.
Finding the sum of the given odd numbers from 15 to 1579
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 15 to 1579
= 783/2 (15 + 1579)
= 783/2 × 1594
= 783 × 1594/2
= 1248102/2 = 624051
Thus, the sum of all terms of the given odd numbers from 15 to 1579 = 624051
And, the total number of terms = 783
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 15 to 1579
= 624051/783 = 797
Thus, the average of the given odd numbers from 15 to 1579 = 797 Answer
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