How do you find the number of terms n given the sum #s_n=1661# of the series #2+9+16+23+30+...#?

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Answer 1 · 2016-11-05T22:12:16

The series has #22# terms.

We know that the common difference is #7#, that the sum of n terms is 1661 and that the first term is #2#.

So, using the formula #s_n = n/2(2a + (n - 1)d)#, we can solve for #n#.

#1661 = n/2(2(2) + (n - 1)7)#

#1661 = n/2(4 + 7n - 7)#

#1661 = n/2(7n - 3)#

#1661 = (7n^2)/2 - (3n)/2#

#3322 = 7n^2 - 3n#

#0 = 7n^2 - 3n - 3322#

#n = (-(-3) +- sqrt(-3^2 - 4 xx 7 xx -3322))/(2 xx 7)#

#n = (3 +- 305)/14#

#n = -302/14 and 308/14->"a negative answer is impossible"#

#n = 22#