Question : If the arithmetic mean of first n natural numbers is 15, then n is equal to
(a) 29
(b) 14
(c) 15
(d) 20
Doubt by Divya
Solution :
Formula Used
Arithmetic Mean (x̄)
= Sum of observations / Total Number of Observations
=(x1+x2+. . . +xn)/n
Formula Used
Arithmetic Mean (x̄)
= Sum of observations / Total Number of Observations
=(x1+x2+. . . +xn)/n
Sum of first n natural numbers
= [n(n+1)]/2
ATQ
x̄ = 15 (Given)
= [n(n+1)]/2
ATQ
x̄ = 15 (Given)
x̄=(x1+x2+. . . +xn)/n
15×n=(x1+x2+. . . +xn)
15n=n(n+1)/2
15=(n+1)/2
30=n+1
30-1=n
n=29
15×n=(x1+x2+. . . +xn)
15n=n(n+1)/2
15=(n+1)/2
30=n+1
30-1=n
n=29
Hence, (a) 29, would be the correct option.