You can start by writing an explicit formula and use it to find the first five terms. Then, try to rewrite it as a recursive formula.
Example sequence: 1,4,9,16,25 Explicit formula: a_n = n^2 for n≥ 1 Recursive formula: a_n = a_(n-1) + (2n-1) for n> 1 and a_1=1
Practice makes perfect
Let's write the following five terms of a sequence.
1, 4, 9, 16, 25
To verify that it is not an arithmetic sequence, we will find the difference between consecutive terms.
There is no common difference, which implies that the sequence is not arithmetic. Next, let's try to find an explicit formula for this sequence.
a_1 &= 1
a_2 &= 4
a_3 &= 9
a_4 &= 16
a_5 &= 25
Notice that each term is equal to the number of its position squared.
a_1 &= 1 = 1^2
a_2 &= 4 = 2^2
a_3 &= 9 = 3^2
a_4 &= 16 = 4^2
a_5 &= 25 = 5^2
From this information we can write the following formula.
Explicit Formula
a_n = n^2 for n≥ 1
To write a recursive formula, let's write the terms as follows.
a_1 &= 1
a_2 &= 4 = 1 + 3
a_3 &= 9 = 4 + 5
a_4 &= 16 = 9 + 7
a_5 &= 25 = 16 + 9
Each term after the first one is equal to the previous term plus a certain odd number. Our next step is to write these odd numbers depending on the position of each term.
a_1 &= 1
a_2 &= 4 = 1 + (2( 2)-1)_3
a_3 &= 9 = 4 + (2( 3)-1)_5
a_4 &= 16 = 9 + (2( 4)-1)_7
a_5 &= 25 = 16 + (2( 5)-1)_9
Following the pattern written above, we can write the following recursive formula.
Recursive Formula
a_1 &= 1
a_n &= a_(n-1) + (2n-1), n > 1
Keep in mind that the sequence used here is just an example, and your answer may vary.
