Final answer:
To find the formula of a sequence, identify the type (arithmetic, geometric, quadratic, or exponential) and use a calculator like the TI-83 or TI-84 to input the sequence data and run the appropriate regression analysis to obtain the formula.
Step-by-step explanation:
To find the formula of a sequence, it's important to recognize the type of sequence we're dealing with. Here are four common types of sequences:
- Arithmetic sequence: Each term is the sum of the previous term and a constant difference (d). Its general formula is an = a1 + (n-1)d.
- Geometric sequence: Each term is the product of the previous term and a constant ratio (r). Its general formula is an = a1rn-1.
- Quadratic sequence: A sequence where the difference between consecutive terms changes at a constant rate. This is often represented as an2 + bn + c.
- Exponential sequence: Each term is a previous term multiplied by a fixed non-zero number raised to the power of the term position. The general formula is an = a1 ⋅ bn, where b is the base and n is the exponent or term number.
To solve for the formula using a calculator, particularly models like the TI-83, TI-83+, or TI-84:
- Input the sequence values as a list.
- Use the sequence's characteristics to decide on the appropriate regression function (e.g., LinReg, QuadReg, ExpReg, or GeoReg for arithmetic, quadratic, exponential, or geometric sequences, respectively).
- Execute the regression to get the formula, which will be displayed in the form of a linear equation for an arithmetic sequence, a quadratic equation for a quadratic sequence, and so on.
By analyzing the data and running the appropriate regression, you can write down the formula for the sequence, rounding to four decimal places if necessary.