Answer:
For a geometric sequence
\(a_1, a_2, a_3, a_4,..\)
The nth term for this sequence is given by:
\(a_n = a_1r^{n-1}\) .....[1]
where
\(a_1\) is the first term
r is the common ratio
n is the number of terms.
Given the sequence:
80, -160, 320, -640
\(a_1 = 80\)
\(a_2 = -160\)
\(a_3 = 320\)
\(a_4= -640\)
Common ratio(r) is -2
Since,
\(r = \frac{a_2}{a_1}=\frac{a_3}{a_2}=\frac{a_4}{a_3}\)
Substitute the values we have;
\(r = \frac{-160}{80}= \frac{320}{-160}=\frac{-640}{320} = -2\)
We have to find the next three term of the given sequence:
Using [1] we have
\(a_5 = a_1 \cdot r^4\)
Substitute the given values we have;
\(a_5 =80 \cdot (-2)^4 = 80 \cdot 16= 1280\)
Similarly,
\(a_6 =80 \cdot (-2)^5= 80 \cdot -32=-2560\)
\(a_7 =80 \cdot (-2)^6 = 80 \cdot 16 =5120\)
Therefore, next three terms in the given sequence are: 1280, -2560, 5120
