Give the explicit form

= 100n - 112

Step-by-step explanation:

Given the recursive formula

= + 100 with a₁ = - 12

This tells us that any term is found by adding 100 to the previous term

That is common difference d = 100 and a₁ = - 12

The explicit formula for an arithmetic sequence is

= a₁ + (n - 1)d , thus

= - 12 + 100(n - 1) = - 12 + 100n - 100 = 100n - 112

there is only one way to arrange all the eight relative to each other because they are not distinct. we have

now there are nine places to place an (and not more than one in one place).   because we have four symbols, we need to choose four of these nine places to insert an . order does not matter because the are not distinct.

thus there are 1 · c(9,4) ways, or, if we calculate it,

thus there are 126 ways to form such "words."