Sequences » Arithmetic progression
In an arithmetic sequence the difference between adjacent terms is always the same.
For an arithmetic sequence with a difference d and initial term t0 is:
The direct formula: tn = t0 + dn.
The recursive formula: tn = tn – 1 + d.
The sum of an arithmetic sequence can be calculated with:
sum = 12 · number of terms · (first term + last term) = 12(n + 1)(t0 + tn)
Given is the following sequence: 43 + 58 + 73 + ... + 358.
|a.||Give the direct formula.|
|b.||Give the recursive formula.|
|c.||Calculate the sum.|
|a.||For the direct formula t0 = 43 and d = 15. The direct formula is: tn = 43 + 15n.|
|b.||The recursive formula for this t0 and d is: tn = tn – 1 + 15.|
|c.||To calculate the sum, we first need to know how many terms are in the sequence.|
The number of terms in the sequence is equal to the number of the last term +1, as you start counting at t0.
To calculate the number of the last term, we need to solve the following equation:
The sum of this sequence is then:
sum = 12 · 22 · (43 + 358) = 4411