Finding Time Intervals: Do the terms in your sequence represent the "time" an event started, or the "time" the event ended? Don't mix them together!

Carl planted a 4 inch sunflower plant. The plant grows 3 inches each week. How tall is the plant at the end of 5 weeks?

The sequence associated with this problem is actually {7, 10, 13, 16, 19},
representing the values at the END of each time interval.

If you use the starting height of 4 inches, with a time of 5 weeks,
you get the WRONG answer. a_n = 4 + 3 (n - 1) = 4 + 3(5 - 1) = 16 inches. NOPE!

The easiest way to address this concern, is to remember to use entries that represent the same thing. 7, 10, 13, ... represent the growth at the END of each time interval, which is asked for in this problem. These are the values we are interested in examining.
The 4 is not a height at the "end" of an interval.

Correct answer: a_n = 7 + 3 (n - 1) = 7 + 3(5 - 1) = 19 inches.

Yes, there are other ways to address this "tick situation". The key is to remember that it needs to be addressed.

Using Simultaneous Equations: You may need to spend some time "finding" the information that appears to be missing.

Find the 30^th term of an arithmetic sequence
where the 5^th term is 30 and the 12^th term is 86.

While we don't know the first term nor the common difference for this sequence
we can set up two equations.
Using a_n = a + (n - 1) d, we have 30 = a + (5 - 1)d and 86 = a + (12 - 1) d.

Simplifying:  30 = a + 4d  and  86 = a + 11d

Two equations with two unknowns means simultaneous equations are an option!

For this specific situation, we saw, in Arithmetic Sequences, that there is another solution option.
If you set up the 5^th term to be a new temporary 1^st term, we can get 86 to be a new 8^th term.
This gives us a first term and the number of terms with which to work.

Using the "patterns" associated with arithmetic (or geometric) sequences: The "patterns" offer a way to represent terms in various locations in the sequences.

(Arithmetic)
Pattern:    a,   a • r,   a • r²,   a • r³,    . . . (Geometric)

The sum of the first four terms of an arithmetic sequence is 28
and the sum of the next four terms is 60.
What is the value of the 25^th term?

This is a "series" problem dealing with "sums".
And we were given very little information regarding this sequence.

While we do not know any specific terms, we do know how to represent them from our "pattern" for an arithmetic sequence.

In the first four terms, a₁ = a and a₄ = a + 3d
In the second four terms, a₅ = a + 4d and a₈ = a + 7d

Use these " expressions" to represent the terms in the formula: