Question

What is the first term in a geometric sequence if the common ratio is −3 and the sum of the first six terms is 1,274?Hint: cap s sub n equals start fraction a sub one (one minus r to the power of n end power )over one minus r end fraction comma r ≠ 1, where a1 is the first term and r is the common ratio.

Answer 1

Answer is   -7

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Work Shown:

s_n = a_1*(1-r^n)/(1-r)

s_6 = a_1*(1-r^6)/(1-r)

s_6 = x*(1-(-3)^6)/(1-(-3))

s_6 = x*(1-729)/(1+3)

s_6 = x*(-728)/4

s_6 = x*(-182)

s_6 = -182x

-182x = s_6

-182x = 1274

x = 1274/(-182)

x = -7 is the first term of the geometric sequence

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Extra info:

The first six terms of this geometric sequence is

-7, 21, -63, 189, -567, 1701

those six terms add to

-7+21+(-63)+189+(-567)+1701 = 1274

which verifies we have the right answer.