Sequences

Jumping to Later Terms of APs

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Jumping to Later Terms of APs
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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.

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Example
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Suppose that an AP has a 45th term of 180 and a 57th term of 228. Find 

 

  1. The 62nd term; and

  2. 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.


  1.  Find the 62nd term.

  

T₆₂ = 4 + (61)(4) = 248.

 

2. Find the first term.

 

T₁ = 4.


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