What Is the Sum of a Linear Number Sequence?
This calculator deals with finding the sum of a linear number sequence — a set of values where every consecutive one differs by the same amount from the previous one. An example would be 20, 23, 26, 29: it's a sequence of 4 numbers, with 20 as the initial value, 29 as the final value, and 3 as the difference between the successive figures. The sum of all of these numbers is 98.
A linear sequence is the same thing as an arithmetic sequence, so the terms "linear number sequence" and "arithmetic sequence" describe exactly the same idea. If you need to quickly find the value of a particular number in a linear number sequence rather than the total, check out our Arithmetic Sequence Calculator.
What Is the Formula for the Sum of a Linear Sequence?
The formula for the sum of a linear sequence reads:
sum = n / 2 × (2 × a + diff × (n − 1))
where:
- a — the initial value of the sequence;
- diff — the difference between any two consecutive numbers in your sequence;
- n — the number of periods (of elements in your sequence); and
- sum — the sum you're looking for.
Let's translate the above formula into a sequence (pun unintended) of easily executable steps!
How Do I Find the Sum of a Linear Sequence?
To compute the sum of a linear sequence:
- Write down the initial value of your sequence a.
- Compute the final value of your sequence as a + diff × (n − 1), where diff is the difference of your sequence and n is the number of its elements.
- Add together the results of Step 1 and Step 2.
- Multiply the result by n.
- Divide the result by 2. That's it!
How Do I Find the Final Value of a Linear Sequence?
To determine the final value of a linear sequence:
- Write down the initial value a of your sequence as well as the difference diff between the two consecutive steps.
- Decide how many elements you want in your sequence. Denote this value by n.
- Compute diff × (n − 1).
- Add a to the number obtained in Step 3. This is your result: the value of the final, i.e., the n-th element of your linear sequence.
A Practical Example
Let's say we're selling storage space for photos of goats on the cloud. Customers pay us an amount of money that's directly proportional to the amount of data they upload — $1 per 1 Gigabyte. Alice initially uploads 5 GB of family photos and then adds an additional 2 GB per month. For the purpose of our analytics, we want to know how much she stores after a year and how much her business is worth to us during that time.
Set the calculator to a = 5, diff = 2, and n = 12 (and pick USD as the currency if you want the dollar sign). Below is the table with all of the values:
| Month | Payment | Sum |
|---|---|---|
| 1 | 5 | 5 |
| 2 | 7 | 12 |
| 3 | 9 | 21 |
| 4 | 11 | 32 |
| 5 | 13 | 45 |
| 6 | 15 | 60 |
| 7 | 17 | 77 |
| 8 | 19 | 96 |
| 9 | 21 | 117 |
| 10 | 23 | 140 |
| 11 | 25 | 165 |
| 12 | 27 | 192 |
After a year Alice stores 27 GB (the final value), and the total value of her uploads over the 12 months is 192 — exactly what the sum formula gives: 12 / 2 × (2 × 5 + 2 × (12 − 1)) = 6 × 32 = 192.
Units and Currencies Supported
Our calculator supports three types of units so you can work in any domain:
- Metric: m, cm, mm, km, kg, g, L, mL, m², m³, km/h
- Imperial (US): ft, in, mi, yd, lb, oz, gal, fl oz, ft², ft³, mph
- Currencies: USD ($), RUB (₽), EUR (€), GBP (£), JPY (¥), CNY (¥), INR (₹), BRL (R$), AUD (A$), CAD (C$), CHF, KRW (₩), MXN (MX$), SEK/NOK/DKK (kr), PLN (zł), TRY (₺), HKD (HK$), SGD (S$)
FAQs
- Are linear sequences and arithmetic sequences the same?
- Yes, the terms linear sequence and arithmetic sequence describe the same type of sequences: those that arise by always adding the same amount to the previous value to get the next value. This amount is called the difference of the linear/arithmetic sequence.
- What is the sum of the first 100 numbers?
- The answer is 5050. This is because the natural numbers form a linear sequence of n = 100 elements with the initial value a = 1 and the difference diff = 1. Applying the formula for the sum of a linear sequence, we get sum = n / 2 × (2 × a + diff × (n − 1)) = 100 / 2 × (2 × 1 + 1 × (100 − 1)). It follows easily that sum = 5050.
- Can the difference be negative or a decimal?
- Absolutely. A negative difference produces a decreasing sequence (e.g., 20, 17, 14, …), and a fractional difference such as 0.5 produces 0, 0.5, 1, 1.5, … The formula works for any real initial value and difference.
- What happens when the difference is zero?
- You get a constant sequence in which every term equals the initial value a. The sum is then simply sum = n × a.