Find the 7$^{th}$ term of the sequence -10, 50, -250 ...........

1. Analyze the question

We are given a sequence: -10, 50, -250, ... and we need to find the 7th term. First, we must identify the type of sequence. By dividing consecutive terms:

• $50 \div (-10) = -5$
• $(-250) \div 50 = -5$

Since there is a common ratio ($r = -5$), this is a Geometric Progression (G.P.).

2. Explain the concept

The formula for the $n^{th}$ term of a Geometric Progression is:

Where:

• $a$ = the first term
• $r$ = the common ratio
• $n$ = the position of the term we want to find

3. Show the solution

From the sequence:

• $a = -10$
• $r = -5$
• $n = 7$

Now, substitute these values into the formula:

Calculate $(-5)^6$: Since the power (6) is an even number, the result will be positive. $(-5)^6 = 5^6 = 15,625$

Now, multiply by the first term ($a$):

4. Identify the answer

Comparing our result to the given options: A. 165250 B. 156250 C. -156250 D. -165,250

The correct option is C.

5. Exam Tip

When dealing with sequences where the ratio is negative, remember the Alternating Sign Rule:

• If the ratio is negative, the signs of the terms will alternate (negative, positive, negative, positive...).
• Since the 1st term is negative, all odd-numbered terms (1st, 3rd, 5th, 7th) will be negative, and all even-numbered terms will be positive. This quick check helps you eliminate options A and B immediately!