Final answer:
To find the sum of the geometric series S₇ for the given sequence, we need to determine the common ratio (-1/3) and plug it into the formula S = a(1 - r^n) / (1 - r). The first term (a) is 162 and the number of terms (n) is 7. Simplifying the expression gives us 162 as the sum.
Step-by-step explanation:
The given sequence is 162, -54, 18, -6,...
To find the sum of the geometric series, we need to determine the common ratio (r) first. The ratio between any two consecutive terms in the sequence is -54/162 = -1/3. Therefore, r = -1/3.
The formula to find the sum of a geometric series is S = a(1 - r^n) / (1 - r), where S is the sum, a is the first term, r is the common ratio, and n is the number of terms.
In this case, a = 162, r = -1/3, and n = 7 (since we need to find S₇). Plugging these values into the formula, we get:
S₇ = 162 * (1 - (-1/3)^7) / (1 - (-1/3))
Simplifying the expression, we get:
S₇ = 54 * (1 - (-1/3)^7) / (4/3)
Finally, we can simplify further to get the answer:
S₇ = 54 * (1 - 1/2187) / (4/3)
S₇ = 54 * (2186/2187) / (4/3)
S₇ = 54 * (1093/1093) / (4/3)
S₇ = 54 / (4/3)
S₇ = 54 * (3/4)
S₇ = 162
Therefore, the indicated sum for the geometric series S₇ is 162.