What is the formula for the sum of an arithmetic sequence?

2 Answers
Jun 27, 2018

# S_n = n/2(2a+(n-1)d) #

Explanation:

Suppose we have an AP with first term #a# and common difference #d#, then we can write the sum of the first #n# terms as:

# S_n = a + (a+d) + (a+2d) + ... + (a+(n-1)d) #

Writing the same sum, but in reverse, we get:

# S_n = (a+(n-1)d) + ... (a+2d) + (a+d) + a #

If we add both of these we get:

# 2S_n = (2a+(n-1)d) + (2a+(n-1)d) + ... + (2a+(n-1)d) #
# \ \ \ \ \ = n(2a+(n-1)d) #

Leading to the standard AP summation formula

# S_n = n/2(2a+(n-1)d) #

Jun 27, 2018

The sum of an arithmetic sequence is given by #S_n=sum_(i=1)^n a_i=n/2(a_1+a_n)#.

Explanation:

Let #S_n# be the sum of the arithmetic sequence and the #n^"th"# term of the sequence be #a_n=a_1+d(n-1)# where #d# is the common difference.
#S_n=a_1+(a_1+d)+(a_1+2d)+...+(a_n-d)+a_n#
#S_n=a_n+(a_n-d)+(a_n-2d)+...+(a_1+d)+a_1#
Adding up the two equations term by term we get
#2S_n=(a_1+a_n)+(a_1+a_n)+...+(a_1+a_n)+(a_1+a_n)#
*Notice how the #d-d# and #2d-2d# all cancel out
Since we have #n# terms of #(a_1+a_n)# on the right side, we can re-write the equation as
#2S_n=n(a_1+a_n)#
#:.S_n=n/2(a_1+a_n)#