Find the sum of:
a) the first odd 'n' odd positive integers
That is the sequence 1, 3, 5, 7, 9, …
a1 = first term = 1
d = common difference = 2nd-1st = 3-1 = 2
formula for sum of first n terms: Sn = [n/2][2a1+(n-1)d]
Sn = [n/2][2a1+(n-1)d]
Sn = [n/2][2∙1+(n-1)2]
Sn = [n/2][2+2(n-1)]
Sn = [n/2][2+2n-2]
Sn = [n/2][2n]
Sn = [n][n]
Sn = n²
b) the first 'n' even positive integers
That is the sequence 2, 4, 6, 8, 10, …
a1 = first term = 2
d = common difference = 2nd-1st = 4-2 = 2
formula for sum of first n terms: Sn = [n/2][2a1+(n-1)d]
Sn = [n/2][2a1+(n-1)d]
Sn = [n/2][2∙2+(n-1)2]
Sn = [n/2][4+2(n-1)]
Sn = [n/2][4+2n-2]
Sn = [n/2][2n+2]
Sn = [n/2][2(n+1)]
Sn = n(n+1)
c) the first 'n' positive integers and find 'n' if the sum is 210.
That is the sequence 1, 2, 3, 4, 5, …
a1 = first term = 1
d = common difference = 2nd-1st = 2-1 = 1
Sn = [n/2][2a1+(n-1)d],
210 = [n/2][2(1)+(n-1)(1)]
210 = [n/2][2+n-1]
210 = [n/2][n+1]
Multiply both sides by 2 to clear the fraction:
420 = n[n+1]
420 = n² + n
0 = n² + n - 420
n² + n - 420 = 0
(n+21)(n-20) = 0
n+21=0; n-20=0
n=-21; n=20
Discard the negative value since the number of terms is always
a positive whole number.
Answer: 20
Edwin