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------------------------------------------------ What is the nᵗʰ term of the sequence 25, -125, 625, -3125, …?

A. (-5)²ⁿ⁻¹
B. (-1)²ⁿ⁵ⁿ⁺¹
C. (-1)²ⁿ – ¹ ⁵ⁿ⁺¹
D. (-1)ⁿ⁻¹ ⁵ⁿ⁺¹

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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