How do you find the sum of the arithmetic series -12-9-6-...+39?

1 Answer
May 9, 2016

Answer:

There are 2 formulae to do this, but you need to find how many terms there are first.

#"S_18 = 243#

Explanation:

The sum of an Arithmetic series can be found from:

#S_n = n/2[2a + (n-1)d]" "# or #" "S_n = n/2(a + l)#

However both of these require knowing how many terms there are.

Each term is defined by #T_n = a + (n - 1)d#

What do we know already?
#a = -12, " " d = 3, " " T_n = 39#

#T_n = (-12) + (n-1)(3) = 39" solve to find n"#

#-12 + 3n - 3 = 39#
#3n = 39 + 15#
#3n = 54#
# n = 18 " there are 18 terms"#

Now we can use either formula for the sum of the first 18 terms.
#S_n = n/2[2a + (n-1)d]" "# or #" "S_n = n/2(a + l)#

#S_18 = 18/2[2(-12) + (18-1)3] "# or #" "S_18 = 18/2(-12 + 39)#

#S_18 = 9[-24 + 51]" "# or #" "S_18 = 9(27)#
#S_18 = 9 xx 27" "# or #" "S_18 = 243#
#"S_18 = 243#

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