Find the 10th Term of Geometric Sequence: 1, -9, 81, ...

Math Problem Statement

1,-9,81,... 10th term

To find the 10th term of the sequence $1, -9, 81, \ldots$ , we first need to determine the pattern in the sequence.

The given sequence is: $1, -9, 81, \ldots$

Looking at the pattern, each term seems to be multiplying the previous term by $-9$ :

\text{First term:} & \quad 1 \\ \text{Second term:} & \quad 1 \times (-9) = -9 \\ \text{Third term:} & \quad -9 \times (-9) = 81 \\ \end{align*}$$ The sequence is a **geometric progression** where the first term $$a = 1$$ and the common ratio $$r = -9$$. ### Step 2: Find the General Formula The general formula for the $$n$$-th term of a geometric sequence is given by: $$a_n = a \cdot r^{n-1}$$ ### Step 3: Calculate the 10th Term To find the 10th term ($$a_{10}$$): $$a_{10} = 1 \cdot (-9)^{10-1} = (-9)^9$$ Now, compute $$(-9)^9$$: $$(-9)^9 = -(9^9)$$ Calculate $$9^9$$: $$9^9 = 387,420,489$$ Thus, $$(-9)^9 = -387,420,489$$ ### Conclusion The 10th term of the sequence is: $$\boxed{-387,420,489}$$ Would you like more details or have any questions? ### Related Questions 1. What would be the 15th term of this sequence? 2. How do you determine if a sequence is geometric or arithmetic? 3. Can you find the sum of the first 10 terms of this geometric sequence? 4. What happens to the sign of the terms in a geometric sequence with a negative common ratio? 5. How would you calculate the $$n$$-th term if the first term is not 1 but some other number? ### Tip Always check the common ratio in a geometric sequence; if it's negative, the signs of the terms alternate.

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