How do you find the number of terms n given the sum #s_n=182# of the series #50+42+34+26+18+...#?

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Answer 1 · 2016-08-20T02:43:44

There are #7# terms in this series.

We use the formula #s_n = n/2(2a + (n - 1)d)# to determine the sum of an arithmetic series.

Here, #d = -8#, #n = ?#, #a = 50# and #s_n = 182#. We will therefore be solving for #n#.

#182 = n/2(2(50) + (n - 1)-8)#

#182 = n/2(100 - 8n + 8)#

#182 = n/2(108 - 8n)#

#182 = (108n)/2 - (8n^2)/2#

#182 = 54n - 4n^2#

#4n^2 - 54n + 182 = 0#

#n = (-(-54) +- sqrt(54^2 - 4 xx 4 xx 182))/(2 xx 4)#

#n = (54 +- sqrt(4))/8#

#n = (54 + 2)/8 and n = (54 - 2)/8#

#n = 7 and n = 6.5#

Since a number of terms that isn't a whole number isn't possible, the aforementioned series has #7# terms.

Hopefully this helps!