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Answer :
Final answer:
The nᵗʰ term of the sequence 25, -125, 625, -3125, … can be represented by the formula (-1)ⁿ⁻¹ ⁵ⁿ⁺¹ (option d). This formula takes into account the alternating sign and the increasing powers of 5 in the sequence.
Explanation:
The nᵗʰ term of the sequence 25, -125, 625, -3125, …… is given by (-1)ⁿ⁻¹ ⁵ⁿ⁺¹. Let's break it down step by step:
- Notice, each term in the sequence is the product of -1 raised to a different power and 5 raised to an increasing power. This is the base structure of our formula.
- Every second term in the sequence is negative, which is accounted for by (-1)ⁿ⁻¹.
- The absolute value of each term in the sequence is a power of 5 that increases with each step. This is represented by the term ⁵ⁿ⁺¹.
Hence, the pattern in this sequence corresponds to the formula (-1)ⁿ⁻¹ ⁵ⁿ⁺¹, which is option d.
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