Пошаговые инструкции
Identify the Given Values
First, identify the given values: the first term (a1) and the common difference (d). These values are crucial in calculating the nth term and partial sum.
Apply the Formula for the nth Term
Next, plug in the values into the formula for the nth term: an = a1 + (n - 1)d. Make sure to substitute the values correctly and perform the calculations carefully.
Calculate the Partial Sum
To calculate the partial sum, use the formula: Sn = (n/2)(a1 + an). First, find the nth term using the formula from step 2, then plug in the values into the partial sum formula.
Check for Common Mistakes
Common mistakes to avoid include incorrect substitution of values, forgetting to subtract 1 from n in the nth term formula, and miscalculating the partial sum. Double-check your calculations to ensure accuracy.
Use a Calculator for Convenience
While it's essential to learn how to calculate the nth term and partial sum manually, you can use a calculator to verify your results or for convenience when dealing with large numbers. Make sure to understand the underlying formulas and calculations before relying on a calculator.
Introduction to Arithmetic Sequences
Arithmetic sequences are a fundamental concept in mathematics, where each term after the first is obtained by adding a fixed constant to the previous term. This constant is called the common difference. In this guide, we will learn how to calculate the nth term and partial sum of an arithmetic sequence manually.
Understanding the Formula
The formula to find the nth term of an arithmetic sequence is given by: an = a1 + (n - 1)d where an is the nth term, a1 is the first term, n is the term number, and d is the common difference.
The formula to find the partial sum of an arithmetic sequence is given by: Sn = (n/2)(a1 + an) where Sn is the partial sum, n is the number of terms, a1 is the first term, and an is the nth term.
Worked Example
Let's consider an example to illustrate the calculation. Suppose we have an arithmetic sequence with a first term of 2 and a common difference of 3. We want to find the 5th term and the partial sum of the first 5 terms.
To find the 5th term, we plug in the values into the formula: a5 = 2 + (5 - 1)3 a5 = 2 + (4)3 a5 = 2 + 12 a5 = 14
To find the partial sum, we first need to find the 5th term, which we already calculated as 14. Then, we plug in the values into the formula: S5 = (5/2)(2 + 14) S5 = (5/2)(16) S5 = 2.5 * 16 S5 = 40
Step-by-Step Guide
Here are the steps to follow: