The given sequence is: -10, 50, -250, .......
To find the pattern, let's first determine if this is a geometric sequence. A geometric sequence has a common ratio between consecutive terms.
The first term \( a_1 \) is -10.
The second term \( a_2 \) is 50.
Let's find the common ratio (r) by dividing the second term by the first term:
\( r = \frac{a_2}{a_1} = \frac{50}{-10} = -5 \)
To confirm it's a geometric sequence, calculate the ratio for the next pair:
The third term \( a_3 \) is -250.
\( r = \frac{a_3}{a_2} = \frac{-250}{50} = -5 \)
The common ratio is confirmed to be **-5**. Therefore, this is a **geometric sequence** with the first term **-10** and a common ratio of **-5**.
The general formula for the n-th term of a geometric sequence is:
\( a_n = a_1 \times r^{(n-1)} \)
To find the 7th term (\( n = 7 \)):
\( a_7 = -10 \times (-5)^{(7-1)} = -10 \times (-5)^6 \)
Compute \( (-5)^6 \):
\(-5 \times -5 \times -5 \times -5 \times -5 \times -5 = 15625 \)
Thus,
\( a_7 = -10 \times 15625 = -156250 \)
The 7th term of the sequence is **-156250**.
The corresponding answer is **-156250**.