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The recursive formula of the explicit rule an = (n + 1)² is [tex]a_n = a_{n-1} + 2n + 1[/tex] where a1 = 4
The explicit formula is given as:
an = (n + 1)²
This means that:
a1 = (1 + 1)² = 4
a₂ = (2 + 1)² = 9
a₃ = (3 + 1)² = 16
Rewrite as:
a₂ = 4 + 2 * 2 + 1 = 9
a₃ = 9 + 2 * 3 + 1 = 16
Substitute a₂ = 9
a₃ = a₂ + 2 * 3 + 1 = 16
Express 2 as 3 - 1
a₃ = a₃₋₁ + 2 * 3 + 1
Express 3 as n
[tex]a_n = a_{n-1} + 2 * n + 1[/tex]
Evaluate the product
[tex]a_n = a_{n-1} + 2n + 1[/tex]
Hence, the recursive formula of the explicit rule an = (n + 1)² is [tex]a_n = a_{n-1} + 2n + 1[/tex] where a1 = 4
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