I am currently reviewing some pre-calculus and I remember the part of sequences and series from taking it in academic format and a trivial question hit me this time around. The known formula for an arithmetic sequence is that the $n^{th}$ term of an arithmetic sequence with first term $b$ and with difference $d$ between consecutive terms is $b + (n-1)d$. Which I get know the explicitly in following the rule when I tried to solve the following example on my own...
Suppose at the beginning of the year your iPod contains 53 songs and that you purchase four new songs each week to place on your iPod. Consider the sequence whose nth term is the number of songs on your iPod at the beginning of the nth week of the year.
(a) What are the first four terms of this sequence?
(b) What is the 30th term of this sequence? In other words, how many songs will
be on your iPod at the beginning of the 30th week?
The book's solution following the formula listed above goes as follows:
solution
(a) The first four terms of this sequence are 53, 57, 61, 65.
(b) To find the 30th term of this sequence, use the formula in the box above
with b = 53, n = 30, and d = 4. Thus at the beginning of the 30th week, the
number of songs on the iPod will be 53 + (30 − 1) · 4, which equals 169.
When I try to solve it on my own I assume that $a_n$ is the number of song at the end of the $n^{th}$ week and my answer for the first four terms is 61,65,69,173 and $a_{30}= 173$ the formula I seem to be following is $a_n=53 + 4n$ which to the best of my knowledge would be the correct answer being that the question says "the beginning of the year your iPod contains 53 songs...four songs each week. and if the initial song count is to be represented in this sequence it would be by $a_0$ not $a_1$. I would appreciate the direction and some feedback regarding by flawed logic if viewed as such. P.S: this is me learning on my own so please do not be considered with "answering my homework" and feel free to provide explicit answers.
Thank you!