Question

Write the algebraic form of the arithmetic sequence got by adding 2 to each term of the above sequence.

Hint:

If you add a constant value to every term of an arithmetic sequence, you get a new arithmetic sequence with the same common difference. Only the first term changes. This can simplify finding the new algebraic form.

Answer 1

We need to create a new sequence by adding 2 to every term of the original sequence (1, 5, 9, 13, ...) and then find the algebraic formula for this new sequence.

The algebraic form (n-th term) of an arithmetic sequence is aₙ = a + (n-1)d. We can either find the formula for the original sequence and add 2, or we can find the new sequence first and then derive its formula.

Method 1: Find the new sequence first.
Original sequence: 1, 5, 9, 13, ...
Add 2 to each term:
New sequence: (1+2), (5+2), (9+2), (13+2), ...
New sequence: 3, 7, 11, 15, ...
For this new sequence:
- First term, a = 3.
- Common difference, d = 7 - 3 = 4.
Now, find its algebraic form:
aₙ = a + (n-1)d aₙ = 3 + (n-1)4 aₙ = 3 + 4n - 4 aₙ = 4n - 1 Method 2: Modify the algebraic form of the original sequence.
The algebraic form of the original sequence (1, 5, 9, ...) is xₙ = 1 + (n-1)4 = 1 + 4n - 4 = 4n - 3.
The new sequence aₙ is obtained by adding 2 to each term of xₙ.
aₙ = xₙ + 2 aₙ = (4n - 3) + 2 aₙ = 4n - 1 Both methods yield the same result.

The algebraic form of the new sequence is 4n - 1.