The goal is to find an explicit formula for \(a_n\), the term in position \(n\), when two terms of a
sequence are known.
1. Arithmetic sequence from two terms
An arithmetic sequence has a constant difference \(d\).
\[
a_n=a_1+(n-1)d
\]
Suppose two terms are known:
\[
a_{n_1}
\qquad
a_{n_2}
\]
The common difference is:
\[
d=\frac{a_{n_2}-a_{n_1}}{n_2-n_1}
\]
After finding \(d\), find the first term:
\[
a_1=a_{n_1}-(n_1-1)d
\]
Then write:
\[
a_n=a_1+(n-1)d
\]
2. Arithmetic example
Suppose:
\[
a_5=17
\qquad
a_8=32
\]
First find the common difference:
\[
d=\frac{32-17}{8-5}
\]
\[
d=\frac{15}{3}=5
\]
Now find \(a_1\):
\[
a_1=17-(5-1)5
\]
\[
a_1=17-20=-3
\]
Therefore:
\[
a_n=-3+(n-1)5
\]
Simplify:
\[
a_n=5n-8
\]
So for \(a_5=17\) and \(a_8=32\), the correct arithmetic formula is:
\[
\boxed{a_n=5n-8}
\]
3. Geometric sequence from two terms
A geometric sequence has a constant ratio \(r\).
\[
a_n=a_1r^{n-1}
\]
Suppose two terms are known:
\[
a_{n_1}
\qquad
a_{n_2}
\]
Divide the two formulas:
\[
\frac{a_{n_2}}{a_{n_1}}=r^{n_2-n_1}
\]
Then solve for \(r\):
\[
r=\left(\frac{a_{n_2}}{a_{n_1}}\right)^{1/(n_2-n_1)}
\]
After finding \(r\), find \(a_1\):
\[
a_1=\frac{a_{n_1}}{r^{n_1-1}}
\]
4. Geometric example
Suppose:
\[
a_2=6
\qquad
a_5=162
\]
Divide the terms:
\[
\frac{162}{6}=r^{5-2}
\]
\[
27=r^3
\]
\[
r=3
\]
Now find \(a_1\):
\[
a_1=\frac{6}{3^{2-1}}=2
\]
Therefore:
\[
\boxed{a_n=2\cdot3^{n-1}}
\]
5. Why the two indices must be different
If \(n_1=n_2\), the two known terms are really at the same position. Then there is not enough information
to find a difference or ratio.
\[
n_2-n_1=0
\]
Division by zero would occur in the arithmetic formula, and the geometric exponent difference would be zero.
6. Visual verification
Once the formula is found, it should reproduce both known terms.
\[
a_{n_1}=\text{first known value}
\]
\[
a_{n_2}=\text{second known value}
\]
The generated table and graph help verify that the formula passes through both known terms.
7. Summary table
| Sequence type |
Use this first |
Then find \(a_1\) |
Final formula |
| Arithmetic |
\(d=\frac{a_{n_2}-a_{n_1}}{n_2-n_1}\) |
\(a_1=a_{n_1}-(n_1-1)d\) |
\(a_n=a_1+(n-1)d\) |
| Geometric |
\(\frac{a_{n_2}}{a_{n_1}}=r^{n_2-n_1}\) |
\(a_1=\frac{a_{n_1}}{r^{n_1-1}}\) |
\(a_n=a_1r^{n-1}\) |
8. Common mistakes
- Do not use \(a_n=3n+2\) for \(a_5=17\), \(a_8=32\). It does not match the second term.
- For arithmetic sequences, subtract term values and divide by the difference of indices.
- For geometric sequences, divide term values and use the difference of indices as the exponent.
- The formula \(a_n=a_1r^{n-1}\) uses exponent \(n-1\), not \(n\).
- Always verify the formula by substituting both known indices.
