Correct Answer - Option 1 : 15
Given:
S = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29}
Formula Used:
For an AP series,
an = a + (n – 1)d
Sn = n(a + an)/2
where, ‘a’ → First term, ‘an’ → Last term, ‘d’ → Common difference, ‘Sn’ → Sum of series, ‘n’ → Number of terms in series.
Arithmetic Mean = Sum of terms/Number of terms
Calculations:
S = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29}
These terms are in AP
Common difference = d = 3 – 1 = 2
First term = a = 1
Last term = an = 29
an = a + (n – 1)d
⇒ 29 = 1 + (n – 1) × 2
⇒ n – 1 = 14
⇒ n = 15
Sn = n(a + an)/2
⇒ Sn = 15 × (1 + 29)/2
⇒ Sn = (15 × 30)/2
⇒ Sn = 225
Arithmetic Mean = 225/15
⇒ Arithmetic Mean = 15
∴ The arithmetic mean of the set S is 15.
Short Trick/Topper’s Approach:
S = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29}
These terms are in AP
Common difference = d = 3 – 1 = 2
First term = a = 1
Last term = an = 29
an = a + (n – 1)d
⇒ 29 = 1 + (n – 1) × 2
⇒ n – 1 = 14
⇒ n = 15
Sum of first n odd numbers = n2
Arithmetic Mean = n2/n = n
⇒ Arithmetic Mean = 15
∴ The arithmetic mean of the set S is 15.