The geometric progression is given by the condition bn = 320⋅ (- 1/2) n. Find b7.

To find the 7th term of a geometric progression, you need to know its first term and the denominator of the progression. To find the denominator, you need to know, in addition to the first term, the second one. Therefore, we find the first second terms of the progression by substituting the numbers 1 and 2 into the formula bn = 320⋅ (- 1/2) n.

b1 = 320 ⋅ (- 1/2) * 1 = 320 * (- 1/2) = – 160;

b2 = 320 ⋅ (- 1/2) * 2 = 320 * (- 1) = – 320.

The denominator is the quotient of the second and first terms of the progression; q = b2 / b1;

q = – 320 / (- 160) = 2.

Find the seventh term of the progression by the formula bn = b1 * q ^ (n – 1).

b7 = – 160 * 2 ^ (7 – 1) = – 160 * 2 ^ 6 = – 160 * 64 = – 10240.

Answer. – 10240.