The General Term Formula
Suppose the first term of an arithmetic sequence is u_{1} and the common difference is d. Then u_{2}=u_{1}+d, u_{n}=u_{1}+2d, u_{4}=u_{1}+3d, and so on. Hence u_{n}=u_{1}+(n-1)d. Term number n, the coefficient of d is one less than the term number (n-1).
For an arithmetic sequence with first term u_{1} and common difference d the general term or nth term is u_{n}=u_{1}+(n-1)d.
Consider this arithmetic sequence: 3, 7, 11, 15, 19, 23, …
To determine an expression for the general term, t_{n}, use the pattern in the terms. The common difference is 4. The first term is 3.
t_{1} | 3=3+4(0) |
---|---|
t_{2} | 3=3+4(1) |
t_{3} | 3=3+4(2) |
t_{4} | 3=3+4(3) |
⋮ | |
t_{n} | 3=3+4(n-1) |
For each term, the second factor in the product is 1 less than the term number.
The second factor in the product is 1 less than n, or n-1.
write:
The General Term of an Arithmetic Sequence
An arithmetic sequence with first term, t_{1}, and common difference, d, is:
t_{1}, t_{1}+d, t_{1}+2d, t_{1}+3d, …
The general term of this sequence is: t_{n}=t_{1}+d(n-1)
(Worked Example) Example 1:
State which of the following are arithmetic sequences by finding the difference between successive terms. For those that are arithmetic, find the next term in the sequence, t_{4} and consequently find the functional definition for the nth term for the sequence, t_{n}.
(a) t: {4, 9, 15, …} (b) t:{-2, 1, 4, …}
Solution:
(a) (1) To check that a sequence is arithmetic, see if a common difference exists.
15-9=6
(2) There is no common difference as 5≠6.
Since there is no common difference the sequence is not arithmetic.
(b) (1) To check that a sequence is arithmetic, see if a common difference exists.
4-1=3
(2) The common difference is 3.
The sequence is arithmetic with the common difference d=3.
(3) The next term in the sequence, t_{4}, can be found by adding 3 to the previous term, t_{3}.
4+3=7
(4) To find the functional definition, write the formula for the nth term of the arithmetic sequence.
=(a–d)+nd
(5) Identify the values of a and d.
(6) Substitute a=-2 and d=3 into the formula and simplify.
t_{n}=3n-5
Example 2. Find the 18th, 23rd and nth terms of the arithmetic progression.
-11, -9, -7, -5, …
Solution:
Here a_{1}=-11, and d=-9+11=2
Thus,
=-11+(17)(2)
=-11+34=23;
a_{23}=a_{1}+(23-1)d
=-11+(22)(2)
=-11+44=33; and
a_{n}=a_{1}+(n-1)d
-11(n-1) (2)
=-11+2n-2=2n-13
Example 3: Write an Equation for the nth Term
Write an equation for the nth term of the arithmetic sequence 8, 17, 26, 35, … .
In this sequence, a_{1}=8 and d=9. Use the nth term formula to write an equation.
Formula for nth term, a_{n}=a_{1}+(n-1)d.
a_{n}=8+(n-1)⋅9
a_{n}=8+9n-9
a_{n}=8+9n-9
An equation is a_{n}=8+9n-9.
Example 4. Find the formula for the nth term of the arithmetic sequence
1) 2, 5, 8, …
2) 107, 98, 89, … .
Solution:
1) Here a=2 and d=3, so
2) Here a=107 and d=-9, so
We can also check whether a given number belongs to a given arithmetic sequence.
Example 5:
a) 2, 4, 6, 8, 10, …
Answer: a_{n}=2n
b) 3, 5, 7, 9, 11, 13, …
I know I need a 2n since each term is 2 apart.
Solve 2(1)-x=3 to figure out the number that goes after the 2n.
Answer: a_{n}=2n+1
c) -3, -1, 1, 3, 5, …
Each term is 2 apart, so again I need a 2n.
Solve 2(1)-x=-3 to figure out what to write after the 2n.
Answer: a_{n}=2n – 5
d) 6, 10, 14, 18, 22, …
Each term is 4 apart, so I need a 4n.
Solve 4(1)+x=6 to figure out what number goes after the 4n.
Answer: a_{n}=4n+2
e) 7, 13, 19, 25, …
Each term is 6 apart, so I need a 6n.
Solve 6(1)+x=7 to figure out what number goes after the 6n.
Answer: a_{n}=6n+1
Example 6: Find a formula for the nth term of the arithmetic sequence with initial term a_{1} and common difference d.
a) a_{1}=4; d=3
I know I need a 3n for the common difference to be 3. I just need to solve for the number after the 3n.
a_{1}=3(1)+x
4=3+x
1=x
Answer: a_{n}=3n+1
b) a_{1}=-2; d=1
I need a 1n, for the common difference of 1. I need to solve for the number after the 1n.
a_{1}=1(1)+x
-2=1+x
-3=x
Answer: a_{n}=1n-3 orjust n-3
c) a_{1}=4; d=-3
I need a -3n, for the common difference of -3. I need to solve for the number after the -3n.
a_{1}=-3(1)+x
4=-3+x
7=x
Answer: a_{n}=-3n+7
d) a_{1}=-2; d=-1
I need a -1n, for the common difference of -1. I need to solve for the number after the -1n.
a_{1}=-1(1)+x
-2=-1+x
-1=x
Answer: a_{n}=-1n-1 or just –n-1
Example 7. Consider the sequence 2, 9, 16, 23, 30, …
a) Show that the sequence is arithmetic.
b) Find a formula for the general term u_{n}.
c) Find the 100th term of the sequence.
d) Is (i) 828 (ii) 2341 a term of the sequence?
solution:
a) 9-2=7
23-16=7
30-23=7
The difference between successive terms is constant. So, the sequence is arithmetic, with u_{1}=2 and d=7.
b) u_{n}=u_{1}+(n-1)d
u_{n}=7n-5
c) If n=100, u_{100}=7⋅100-5=700-5=695
d) (i) Let u_{n}=828
7n=833
n=119
∴ 828 is a term of the sequence, and in fact is the 119th term.
(ii) Let u_{n}=2341
7n=2346
n=335+1/7
But it must be an integer, so 2341 is not a member of the sequence.
Example 8. Find the general term u_{n} for an arithmetic sequence with u_{3}=8 and u_{8}=-17.
Solution:
Using u_{n}=u_{1}+(n-1)d.
u_{8}=-17∴u_{1}+7d=-17…(2)
Eliminate u_{1} to get d by (1)-(2).
(1)…u_{1}+2(-5)=8→u_{1}=8+10=18
Now u_{n}=u_{1}+(n-1)d.
u_{n}=18-5n+5
u_{n}=23-5n
Check:
u_{8}=23-5⋅8=23-40=-17✓
Let’s read post Write an equation or explicit formula for the nth term of each geometric sequence.
Example 9. Fill in the blanks in the following table, given that a is the first term, d the common difference and a_{n} the nth term of the AP:
a | d | n | a_{n} | |
---|---|---|---|---|
(i) | 7 | 3 | 8 | … |
(ii) | -18 | … | 10 | 0 |
(iii) | … | -3 | 18 | -5 |
(iv) | -18.9 | 2.5 | … | 3.6 |
(v) | 3.5 | 0 | 105 | … |
Solution:
(i) Here, a=7, d=3 and n=8, so, putting the values in a_{n}=a+(n-1)d, we get:
⇒a_{n}=7+21=28
(ii) Here, a=-18, n=10 and a_{n}=0, so, putting the values in a_{n}=a+(n-1)d, we get:
⇒18=9d ⇒d=2
(iii) Here, d=-3, n=18 and a_{n}=-5, so, putting the values in a_{n}=a+(n-1)d, we get:
–
⇒-5=a-51 ⇒a=46
(iv) Here, a=-18.9, d=2.5 and a_{n}=3.6, putting the values in a_{n}=a+(n-1)d, we get:
⇒3.6=-18.9+2.5n-2.5
⇒2.5n=3.6+21.4=25.0
⇒n=10
(v) Here, a=3.5, d=0 and n=105, putting the values in a_{n}=a+(n-1)d, we get