Is there a common difference? Write each term depending on the previous term. Notice that the number added to the previous term equals the position of the term.
$104
Practice makes perfect
Let's begin by writing the first four payments and the difference between consecutive terms.
Let's try to write a formula that describes the given sequence.
a_1 &= 50
a_2 &= 52 = 50 + 2
a_3 &= 55 = 52 + 3
a_4 &= 59 = 55 + 4
From the above we can see that each term after the first term is obtained by adding the position it occupies in the sequence to the previous term. Thus, we can write the following formula for the given sequence.
a_n = a_(n-1) + n, n> 1, a_1 = 50
To determine how much we will receive in the tenth month, we have to find a_(10). Notice that to find it we need to find all of the previous terms as well.
From the above we conclude that in the tenth month we will receive $104.
B. Identifying the Formula
As we can see in the formula we wrote, each term is related to the previous term and we need to know the first term. Thus, our formula is a recursive formula.
C. Explaining Differences
To find a term in a recursive formula we need to know the previous terms. As we did above, to find a_(10) we had to find all the terms before it.
a_n = a_(n-1) + n, n > 1, a_1 = 50
In contrast, to find a term using an explicit formula we need to know only the position of the term.
Extra
Finding the Explicit Formula
To find an explicit formula for the given sequence, we start by rewriting the first term as 49+1.
a_1 &= 50 = 49 + 1
a_2 &= 52 = 50 + 2
a_3 &= 55 = 52 + 3
a_4 &= 59 = 55 + 4
Next, let's substitute a_1 into a_2.
a_2 &= (49+1) + 2
&= 49 + (1+ 2)
Similarly, let's substitute a_2 into a_3.
a_3 &= (49+1+2) + 3
&= 49 + (1+2+ 3)
Let's do the same process one more time.
a_4 &= (49+1+2+3) + 4
&= 49 + (1+2+3+ 4)
We can then write the following formula.
a_n = 49 + (1+2+3+⋯ + n)
Finally, we have that the sum of the first n positive integers is given by n(n+1)2.
Explicit Formula
a_n = 49 + n(n+1)/2
Using this new formula, let's find a_(10) by substituting n=10.
