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Math  /  Algebra

Question2)
Consider the sequence 1,−23,49,−827,…1,-\frac{2}{3}, \frac{4}{9},-\frac{8}{27}, \ldots. Determine s8s_{8}.

Studdy Solution

STEP 1

1. The sequence is geometric.
2. The first term of the sequence is a=1 a = 1 .
3. The common ratio r r can be determined from the sequence terms.
4. The formula for the n n -th term of a geometric sequence is sn=a⋅rn−1 s_n = a \cdot r^{n-1} .

STEP 2

1. Identify the common ratio r r .
2. Use the formula for the n n -th term of a geometric sequence to find s8 s_8 .

STEP 3

Identify the common ratio r r by dividing the second term by the first term:
r=−231=−23 r = \frac{-\frac{2}{3}}{1} = -\frac{2}{3}
Verify by checking the ratio between subsequent terms:
r=49−23=−23 r = \frac{\frac{4}{9}}{-\frac{2}{3}} = -\frac{2}{3} r=−82749=−23 r = \frac{-\frac{8}{27}}{\frac{4}{9}} = -\frac{2}{3}
The common ratio r r is consistent.

STEP 4

Use the formula for the n n -th term of a geometric sequence to find s8 s_8 :
The formula is:
sn=a⋅rn−1 s_n = a \cdot r^{n-1}
Substitute a=1 a = 1 , r=−23 r = -\frac{2}{3} , and n=8 n = 8 :
s8=1⋅(−23)8−1 s_8 = 1 \cdot \left(-\frac{2}{3}\right)^{8-1} s8=(−23)7 s_8 = \left(-\frac{2}{3}\right)^7
Calculate (−23)7 \left(-\frac{2}{3}\right)^7 :
(−23)7=−2737=−1282187 \left(-\frac{2}{3}\right)^7 = -\frac{2^7}{3^7} = -\frac{128}{2187}
The value of s8 s_8 is:
−1282187 \boxed{-\frac{128}{2187}}

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