Question

Find the first five terms of the given recursively defined sequence. aₙ=aₙ₋₁/2 and a₁=-8

Answer 1

The first five terms of the recursively defined sequence
`\( a_n = (a_(n-1))/(2) \) with \( a_1 = -8 \) are \(-8, -4, -2, -1, -(1)/(2)\).`

Step-by-step explanation:

The recursive formula
`\( a_n = (a_(n-1))/(2) \)` indicates that each term is obtained by dividing the previous term by 2. Given the initial value
`\( a_1 = -8 \),` we can calculate the subsequent terms.

1.
`\( a_2 = (a_1)/(2) = (-8)/(2) = -4 \)`

2.
`\( a_3 = (a_2)/(2) = (-4)/(2) = -2 \)`

3.
`\( a_4 = (a_3)/(2) = (-2)/(2) = -1 \)`

4.
`\( a_5 = (a_4)/(2) = (-1)/(2) = -(1)/(2) \)`

Therefore, the first five terms of the sequence are
`\(-8, -4, -2, -1, -(1)/(2)\).`

This sequence demonstrates exponential decay since each term is half of the preceding term.

As the terms progress, the values get closer to zero, showing the characteristic behavior of a recursively defined sequence with a division factor of 2.