Question

Find the next three terms (a₂, a₃, and a₄) of the following sequence:

a₁ = 2, and for all integers k > 1, ak = aᴋ₋₁ - 3k.
a. Prove using mathematical induction that1/(1×2)+1/(2×3)+⋯+1/n(n+1) =n/(n+1) forallintegersn≥1.

Answer 1

The next three terms of the sequence given by the recursive formula aₖ = aₖ₋₁ - 3ₖ are a₂ = -4, a₃ = -13, and a₄ = -25.

Step-by-step explanation:

Finding the Next Terms

To find the next three terms (a₂, a₃, and a₄) of the given sequence: a₁ = 2, and aₖ = aₖ₋₁ - 3ₖ for all integers k > 1, we will apply the recursive formula.

1. For a₂, we substitute k = 2 into the formula to get a₂ = a₁ - 3(2) = 2 - 6 = -4.
2. To find a₃, use k = 3: a₃ = a₂ - 3(3) = -4 - 9 = -13.
3. For a₄, with k = 4: a₄ = a₃ - 3(4) = -13 - 12 = -25.

Therefore, the next three terms are a₂ = -4, a₃ = -13, and a₄ = -25.