10.3
Rekenkundige rijen
An arithmetic sequence is a list of numbers where each term increases or decreases by the same fixed number, known as the common difference. Consider a pile of poles. The first layer contains 25 poles, and the number of poles continues to decrease by 1 in each successive layer.
Given that the pile has 12 layers, the goal is to find the total number of poles.
This arrangement forms an arithmetic sequence, as the number of poles decreases by a constant amount from one layer to the next.
In this scenario, the number of poles in the 12th layer is calculated using the formula for the nth term of an arithmetic sequence, based on the first term, the common difference, and the number of layers. The values of these terms are then substituted into the formula, which simplifies to 25 minus 11, giving 14 poles in the 12th layer.
The total number of poles in the pile, known as the partial sum of the sequence, is then calculated by taking the average of the number of poles in the first and last layers and multiplying it by the total number of layers. It is termed a partial sum since only the first 12 terms of the sequence are added. This results in a partial sum of 12 multiplied by the average of 25 and 14, yielding 234 poles.
Een rekenkundige rij is een opeenvolging van getallen waarbij elke term ontstaat door een constante waarde, het verschil d, op te tellen bij de voorgaande term. Dit vaste patroon maakt het mogelijk om efficiënt elke gewenste term én de som van meerdere opeenvolgende termen te berekenen. De formule voor de n-de term luidt:
Hierin is a_n de n-de term van de rij, a de eerste term, d het (constante) verschil en n de index (rangnummer) in de rij. Met deze vergelijking kan men elke term bepalen zonder alle voorafgaande termen expliciet te noteren. Voor de som van de eerste n termen (de partialsom) gebruikt men een van de volgende, equivalente formules:
In deze notatie staat S_n voor de som van de eerste n termen, en verwijst a_n opnieuw naar de n-de term zoals hierboven gedefinieerd. Deze formules bieden een compacte en systematische methode om regelmatig gespreide numerieke patronen te analyseren in zowel theoretische als praktische toepassingen.
An arithmetic sequence is a list of numbers where each term increases or decreases by the same fixed number, known as the common difference. Consider a pile of poles. The first layer contains 25 poles, and the number of poles continues to decrease by 1 in each successive layer.
Given that the pile has 12 layers, the goal is to find the total number of poles.
This arrangement forms an arithmetic sequence, as the number of poles decreases by a constant amount from one layer to the next.
In this scenario, the number of poles in the 12th layer is calculated using the formula for the nth term of an arithmetic sequence, based on the first term, the common difference, and the number of layers. The values of these terms are then substituted into the formula, which simplifies to 25 minus 11, giving 14 poles in the 12th layer.
The total number of poles in the pile, known as the partial sum of the sequence, is then calculated by taking the average of the number of poles in the first and last layers and multiplying it by the total number of layers. It is termed a partial sum since only the first 12 terms of the sequence are added. This results in a partial sum of 12 multiplied by the average of 25 and 14, yielding 234 poles.
From Chapter 10:
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