How to Calculate a Term in an Arithmetic Sequence, Given Two Terms?

Arithmetic Sequences

The simplest type of numerical sequence is an arithmetic sequence.

Definition: Arithmetic Sequence
An arithmetic (or linear) sequence is a sequence of numbers in which each new term is calculated by adding a constant value to the previous term

For example, 1, 2, 3, 4, 5, 6, … is an arithmetic sequence because you add 1 to the current term to get the next term:

first term: 1
second term: 2=1+1
third term: 3=2+1

nth term: n=(n-1)+1

General Equation for the nth-term of an Arithmetic Sequence
More formally, the number we start out with is called a1 (the first term), and the difference between each successive term is denoted d, called the common difference.

The general arithmetic sequence looks like:

a1=a1
a2=a1+d
a3=a2+d=(a1+d)+d=a1+2d
a4=a3+d=(a1+2d)+d=a1+3d

an=a1+d⋅(n-1)

Thus, the equation for the nth-term will be:
an=a1+d⋅(n-1)

Given a1 and the common difference, d, the entire set of numbers belonging to an arithmetic sequence can be generated.

Definition: Arithmetic Sequence
An arithmetic (or linear) sequence is a sequence of numbers in which each new term is calculated by adding a constant value to the previous term:

an=a(n-1)+d

where
an represents the new term, the nth-term, that is calculated;
a(n-1)represents the previous term, the (n-1)th-term;
d represents some constant.

Arithmetic Progression (A.P.)

An arithmetic progression is a sequence of terms such that the difference between any term and the one immediately preceding it is a constant. This difference is called the common difference.

For example, the sequences
(i) 3, 7, 11, 15, 19, …
(ii) 7, 5, 3, 1, -1, …
(iii) 1, ¾, ½, ¼, 0, -¼, …
(iv) 2, 2, 2, 2, …
are arithmetic progressions.
In (i) common difference is 4,
In (ii) common difference is -2,
In (iii) common difference is -¼, and
In (iv) common difference is 0,
In general, an arithmetic progression (A.P.) is given by

a, a+d, a+2d, a+3d, ….

We call a as first term and d as the common difference.
The nth term of the above A.P. is denoted by an and is given by an
an=a+(n-1)d

nth term or General term (or, last term) of an Arithmetic Progression:
If ‘a’ be the first term and ‘d’ be the common difference then

a1= first term =a=a+(1-1)d
a2= 2nd term =a+d=a+(2-1)d
a3= 3rd term =a+2d=a+(3-1))d
a4= 4th term =a+3d=a+(4-1)d

an= nth term =a+(n-1)d

an=a+(n-1)d


in which a= 1st term, d= common difference, n= number of terms

Example 1: Calculating a Term in an Arithmetic Sequence, Given Two Terms
Two terms in an arithmetic sequence are t3=4 and t8=34. What is t1?
solution:
t3=4 and t8=34.
Sketch a diagram. Let the common difference be d.

common difference d

From the diagram,
t8=t3+5d

Substitute: t8=34, t3=4
34=4+5d

Solve for d.
30=5d
d=6

Then, t1=t3-2d
Substitute: t3=4, d=6
t1=4-2(6)
t1=4-12
t1=-8

Example 2: Finding a term in an arithmetic sequence
(A) If the first and tenth terms of an arithmetic sequence are 3 and 30, respectively, find the fiftieth term of the sequence.
Solution:
(A) First use Theorem 1 with a1=3 and a10=30 to find d:

an=a1+(n-1)d
a10=a1+(10-1)d
30=3+9d
d=3

Now find a50
a50=a1+(50-1)⋅3
3+49⋅3=150

Example 3. Find the 16th and nth terms in an arithmetic sequence with the fourth term 15 and eighth term 37.
Steps:
(1) Write the formula for the nth term of the arithmetic sequence.

tn=a+(n-1)×d

(2) Substitute n=4 and t4=15 into the formula.
t4=a+3d=15

(3) Substitute n=8 and t8=37 into the formula
t8=a+7d=37

(4) Solve the simultaneous equations: subtract t4 from t8 to eliminate a
(t8t4)…a+7d-a-3d=37-15
4d=22→d=5½

(5) Substitute d=5½ into t4 and solve for a.
(t4)…a+3×5½=15
a=-1½

(6) To find the nth term of the arithmetic sequence, substitute the values of a and d into the general formula and simplify.
tn=-1½+(n-1)×5½
tn=-1½-5½+5½n
tn=5½n-7

(7) To find the 16th term, substitute n=16 into the formula established in the previous step and evaluate.
t16=5½×16-7=88-7
t16=81

Example 4. Find the missing terms in this arithmetic sequence: {41, a, 55, b, …}.
Steps:
(1) The first three successive terms are 41, a, 55. Write the rule for the middle term of the three successive terms of an arithmetic sequence.
For x, y, z: y=½(x+z).
(2) Identify the variables.

x=41; y=a; z=55

(3) Substitute the values of x, y and z into the formula in step 1 and evaluate.
a=½(41+55)=48

(4) Find the common difference. (The second term is now known.)
d=t2t1=48-41=7

(5) Find the value of b by adding the common difference to the preceding term.
b=55+7=62

(6) State your answer.
so a=48, b=62

Example 5. The 4th term of an arithmetic sequence is 80 and the 12th term is 32.
a. What is the common difference?
b. What is the first term of the sequence?
solution:
a. How to Proceed
(1) Use an=a1+(n-1)d to write two equation in two variables:

get common difference by eliminating

(2) Subtract to eliminate a1:
(3) Solve for d:
b. Substitute in either equation to find a1.
80=a1+3(-6)
80=a1-18
98=a1

answer: a. d=-6 b. a1=98

Example 6. Find the 31st term of an AP whose 11th term is 38 and the 16th term is 73.
Solution:
Here, a11=38 and a16=73 to find a31.
Given that: a11=a+(11-1)d=38

a+10d=38
a=38-10d…(1)
and a16=73 ⇒a+15d=73

Putting the value of a from equation (1), we get 38-10d+15d=73
⇒5d=35
d=7

Putting the value of d in equation (1), we get
a=38-10(7)=-32

Therefore, a31=a+30d=-32+30(7)=178
Hence, the 31st term is 178.

Example 7. An AP consists of 50 terms of which 3rd term is 12 and the last term is 106. Find the 29th term.
Solution:
Here, a3=12 and a50=106 to find a29.
Given that: a3=a+(3-1)d=12

a+2d=12
a=12-2d …(1)

and a50=106
a+49d=106

Putting the value of a from equation (1), we get
12-2d+49d=106
47d=94
d=2

Putting the value of d in equation (1), we get
a=12-2(2)=8

Therefore, a29=a+28d=8+28(2)=64
Hence, the 29th term of the AP is 64.

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