Answer:
a₁ = 7
a
= -3 ⋅ a
, for n > 1
Explanation:
Do you want to know how to write a recursive formula for a_n, the n^th term of the sequence: 7, -21, 63, -189...? Well, it's easy as pie! Just follow these simple steps:
1. Find the common ratio of the sequence. This is the number that you multiply each term by to get the next term. In this case, it's -3. You can check by dividing any term by the previous term. For example, -21 / 7 = -3, 63 / -21 = -3, and so on.
2. Write the formula for a
using the common ratio and the previous term. The formula is a
= -3 ⋅ a
), where n is any positive integer greater than 1. This means that to find any term in the sequence, you just multiply the previous term by -3.
3. Write the formula for a₁, the first term of the sequence. This is the starting point of the recursion. In this case, it's 7. You can find it by looking at the first term in the sequence or by plugging in n = 1 into the formula for a
.
4. Voila! You have written a recursive formula for a
. The formula is:
a₁ = 7
a
= -3 ⋅ a
, for n > 1
Congratulations! You are now a master of recursion! Give yourself a pat on the back and celebrate with some pi!