Construct an arithmetic sequence Construct an arithmetic sequence and find the first term if the common difference is 2 and the third term plus the fourth term = 8?
As you have given a common difference we know that it is an AP.
If the 1st term of an AP is a and the common difference is d, the nth term is given as a + (n - 1)d
The third term is a + 2d and the fourth term is a + 3d
We have a + 2d + a + 3d = 8 and d = 2
=> 2a + 10 = 8
=> a = -1
So the required series is -1, 1, 3, 5, 7...
To create an arithmetic sequence, we'll have to know 2 basic terms: the 1st term and the common difference. The common difference is known but the 1st term is not known.
From enunciation we'll get the information that:
a3 + a4 = 8 (1)
By definition, the difference between 2 consecutive terms of an arithmetial progression is the common difference of the arithmetic sequence.
a4 - a3 = d
But, the common difference is d = 2, then:
a4 - a3 = 2 (2)
We'll add (1) + (2):
a3 + a4 - a3 + a4 = 8 + 2
We'll eliminate and combine like terms:
2a4 = 10
a4 = 10/2
a4 = 5
a3 = a4 - 2
a3 = 5 - 2
a3 = 3
But a3 = a1 + 2d
3 = a1 + 4
a1 = 3 - 4
a1 = -1
The first term of the created arithmetic sequence, whose common difference is d=2, is a1 = -1.