All arithmetic sequences have some common difference, d. Using this common difference, and the value of the first term, a1, it's possible to find an explicit rule that describes the sequence. By thinking of the terms in a sequence using a1 and d, a pattern emerges.
|n||Using a1 and d|
When n increases by 1, the coefficient of d increases by 1 as well. Due to this, and that the coefficient is 0 when n is 1, the coefficient is always 1 less than n. Expressing this in a general form gives the explicit rule.
To find the twelfth term of the sequence, we can now substitute n=12 into the rule.
The twelfth term is 37.
Pelle is an avid collector of pellets. During his spare time, he likes to arrange his pellets in different patterns. Today, he's chosen to place them in the shape of a triangle. The top row consists of one pellet, the second of three pellets, the third of five pellets, and so on. Write a rule where is the amount of pellets in row n. Then, use the rule to find which row has 53 pellets.
To begin, we can make sense of the given information. For every row, the amount of pellets increase by 2. Thus, we know that d=2. It is also given that the first row consists of 1 pellet, a1=1. Using this information, we can find the rule.
Row 27 has 53 pellets.
To begin, we must determine the common difference, d. Since we do not know the values of two consecutive terms, we cannot directly find d. However, the terms a3 and a6 are 3 positions apart, so they must differ by 3d.