HOW TO FIND THE NTH TERM OF A SEQUENCE

Write the nth term of the following sequences ;

Example 1 :

2, 2, 4, 4, 6, 6, ........

Solution :

By observing the given sequence first, second terms are same, third and fourth terms are same and so on.

To find nth term of the sequence, let us divide them into two parts.

Odd Terms

Let tn = n + 1

If n = 1

 t1  =  1 + 1  ==> 2

If n = 3

 t3  =  3 + 1  ==> 4

If n = 5

 t5  =  5 + 1  ==> 6

Even Terms

Let tn = n

If n = 2

 t2 =  2

If n = 4

 t4 =  4

If n = 6

 t6 =  6

Hence tn = n + 1  (if n is odd)

               = n (if n is even)

Example 2 :

1/2 , 2/3 , 3/4 , 4/5 , 5/6, ........

Solution :

By observing the given sequence, the numerator is 1 lesser than the denominator.

Let tn  =  n/(n+1) 

t1  =  1/(1+1)  ==>  1/2

t2  =  2/(2+1)  ==>  2/3

t3  =  3/(3+1)  ==>  3/4

Hence the required nth term of the given sequence is n/(n+1).

Example 3 :

1/2, 3/4, 5/6, 7/8, 9/10, ........

Solution :

By observing the given sequence, the numerator is 1 lesser than the denominator and they are odd terms.

Let tn  =  (2n-1)/2n

t1  =  (2(1) - 1)/2(1) ==>  1/2

t2  =  (2(2) - 1)/2(2) ==>  3/4

t3  =  (2(3) - 1)/2(3) ==>  5/6

Hence the required nth term of the given sequence is (2n-1)/2n.

Example 4 :

6, 10, 4, 12, 2, 14, 0, 16, −2, ........

Solution :

By observing the given sequence first, second terms are same, third and fourth terms are same and so on.

To find nth term of the sequence, let us divide them into two parts.

Odd Terms

Let tn = 7 - n

If n = 1

 t1  =  7 - 1  ==> 6

If n = 3

 t3  =  7 - 3 ==> 4

If n = 5

 t5  =  7 - 5  ==> 2

Even Terms

Let tn = 8 + n

If n = 2

 t2 =  8 + 2  ==> 10

If n = 4

 t4 =  8 + 4  ==>  12

If n = 6

 t6 =  8 + 6  ==> 14

Hence the nth term is 

tn = 7 - n  (if n is odd) 

    = 8 + n  (if n is even)

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