Sequences
Jumping to Later Terms of APs
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Jumping to Later Terms of APs
If we used the same thought process that we use for simple sequences, we would recall that sequences follow the below progression
a, a+d, a+2d, a+3d, a+4d, a+5d, …
Note that for T₂ the common difference, d had been added once, for T₃ it had been added twice, for T₄ it has been added three times, and so on.
Knowing this, we can conclude that arithmetic progressions have an nth term given by the rule
Tₙ = a + (n-1)d
Where
a = first term; and
d = common difference.
This format of the rule is known as the nth term rule.
Example
Suppose that an AP has a 45th term of 180 and a 57th term of 228. Find
The 62nd term; and
The 1st term.
Solution:
Firstly, create a pair of simultaneous equations using the information given.
180 = a + 44d
228 = a + 56d
Now, solve the equations to find the value of d.
-48 = -12d
d = 4
Substitute the value of d into one of the original equations to find the value of a.
180 = a + 44(4)
a = 4.
We should now use this information to find solve the question.
Find the 62nd term.
T₆₂ = 4 + (61)(4) = 248.
2. Find the first term.
T₁ = 4.