Question

2.) A farmer mistakenly makes a deal with a local grocery store to give them 1 ear of corn on day 1, 2 ears of corn on day 2, 4 ears of corn on day 3, until the corn doubles for an entire month. Write a recursive formula for this situation.

Answer 1

`\begin{cases}a_1=1\\a_(n+1)=2 a_n\end{cases}`

Explanation:

A recursive formula for a sequence allows you to find the nth term of the sequence provided you know the value of the previous term in the sequence.

The first three terms of the sequence are:

• 
  `a_1=1`

• 
  `a_2=2`

• 
  `a_3=4`

Therefore, as each term of the sequence is twice the previous term it is a geometric sequence.

When writing a recursive rule, remember to include the definition of the first term of the sequence.

Therefore, the recursive formula for the given situation is:

`\begin{cases}a_1=1\\a_(n+1)=2 a_n\end{cases}`

-----------------------------------------------------------------------------------------------

Additional information

The "a" signifies "term". So a₁ is the first term, a₂ is the second term ... aₙ is the nth term. Therefore, aₙ₊₁ is the next term.

"u" is regularly used instead of "a", but as "a" is used as the first term when writing explicit formulas, it is more preferable.

As a recursive formula for a sequence allows you to find the nth term of the sequence provided you know the value of the previous term in the sequence, to find the 6th term for example, we would need to calculate the previous terms first:

`\implies a_1=1`

`\implies a_2=2 \cdot a_1=2 \cdot 1=2`

`\implies a_3=2 \cdot a_2=2 \cdot 2=4`

`\implies a_4=2 \cdot a_3=2 \cdot 4=8`

`\implies a_5=2 \cdot a_4=2 \cdot 8=16`

`\implies a_6=2 \cdot a_5=2 \cdot 16=32`

Note: It is more preferable to find an explicit formula for the nth term of a sequence since with an explicit formula we do not need to calculate the previous term(s).

The explicit formula for this sequence of this question is:

`a_n=2^(n-1)`

Therefore, to find the 6th term using the explicit formula, simply substitute n = 6 into the formula:

`\implies a_6=2^(6-1)=2^5=32`

As you can see, it yields the same result as the recursive formula but without the need to calculate the preceding terms.