Question

Given sequence :2;6;10;14;18;.....
Show that Tn=4n-2 by generating an algebraic expression for the n^th term.

Answer 1

To find the algebraic expression for the nth term (Tn) of the given sequence, we can observe that each term is obtained by adding 4 to the previous term.

Starting with the first term (T1 = 2), the difference between consecutive terms is 4.

T2 = T1 + 4 = 2 + 4 = 6
T3 = T2 + 4 = 6 + 4 = 10
T4 = T3 + 4 = 10 + 4 = 14
T5 = T4 + 4 = 14 + 4 = 18

From this pattern, we can see that each term Tn is obtained by adding 4 to the previous term, and we can express this algebraically.

Tn = T(n-1) + 4

To find the general formula for the nth term, we start with the initial term T1.

T1 = 2

Now we substitute T1 into the formula for Tn:

Tn = T(n-1) + 4

T2 = T(2-1) + 4 = T1 + 4 = 2 + 4 = 6

T3 = T(3-1) + 4 = T2 + 4 = 6 + 4 = 10

T4 = T(4-1) + 4 = T3 + 4 = 10 + 4 = 14

T5 = T(5-1) + 4 = T4 + 4 = 14 + 4 = 18

We can see that the formula Tn = T(n-1) + 4 is valid for all the terms of the sequence.

To further simplify the expression, notice that the initial term T1 can be written as Tn = 4(n-1) + 2.

Therefore, the algebraic expression for the nth term (Tn) of the given sequence is Tn = 4n - 2.