Find the average of all odd numbers up to 100 (1) 30 (2) 50 (3) 4 (4) 10
Answers
SOLUTION :-
=>Odd numbers are defined as any number which cannot be divided by two. In other words, a number of form 2k+1, where k ∈ Z (i.e. integers) are called odd numbers.
=>There are 50 odd numbers from 1 to 100.
=>From 1 to 100 sum of the odd numbers are = 1+3+5+7+9+….+93+95+97+99
=>We can solve the question by Arithmetic Progression(AP) formula
=>Let us consider,
=>a = 1
=>l = 99
=>d = 2
=>T(n) = a + (n – 1)d
=>99 = 1 + (n – 1)2
=>98 = 2n – 2
=>100 = 2n
=> n = 50
=>Now consider,
=>S(n) = n/2(a + l)
=>S(50) = 50/2 (1 + 99)
=>S(50) = 2500
=>Avaregae sum = Sum of numbers / Total number
=>Avaregae sum = 2500/50
=>.°. Avaregae sum = 50.
Step-by-step explanation:
Average = Sn/n
Average =(1+3+5+7+9+…………+99)/n
This is an A.P. in which 1st term (a) = 1 and
n th term tn = 99.
Average =(n/2).(a+tn)/n = (a+ tn)/2.
Average = (1+99)/2 = 100/2 = 50 .Answer.
ODD NUMBERS BETWEEN 0 TO 100 ARE
1,3,5,7……..,99
HERE THESE ARE IN AP
WHERE a = 1
d = 2
So An = a + (n-1)d
99 = 1 + (n-1) 2
98 = (n-1)2
n-1 = 49 => n = 50
Then
Sn = (n/2) * (a + An)
=( 50/2 )* (1+99)
= 25 * 100
= 2500
Average = Sn/ n
= 2500 / 50
= 50