How do you find the number of terms n given the sum #s_n=-12# of the series #34+31+28+25+22+...#?

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Answer 1 · 2016-07-27T09:04:35

There are 24 terms. #n# = 24.

First identify what we already DO know.

It is an AP - the pattern is subtract 3 each time.

#T_1 = a = 34#
#d = -3 " (subtract 3)"#
#"sum" = -12#

The formula for the sum of #n# terms is

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

We do NOT know the number of terms, #n#, but we know all the other values, so just substitute them into the formula.

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

#-12 = n/2[2(34) + (n-1)(-3)] " now solve for n"#

#-12 = n/2[68-3n+3]#

#-24 = n[71-3n]#

#-24 = 71n -3n^2#

#3n^2 -71n - 24 =0#

#(n -24)(3n +1)=0#

#n = 24 or n= -1/3" reject "-1/3 " as the number of terms"#

There are 24 terms.