How to find the first term in an arithmetic sequence?

Understand the Problem

The question is asking for a method to determine the first term of an arithmetic sequence, which can be done using the formula for the nth term of an arithmetic sequence. Typically, this involves knowing the common difference and another term in the sequence.

Answer

Use the formula $a_1 = a_n - (n-1) imes d$

Steps to Solve

Identify the known values in the problem

We need to know the common difference ($d$), the position of the known term ($n$), and the value of the known term ($a_n$). This information allows us to apply the formula for the nth term of an arithmetic sequence.

Write the formula for the nth term

The formula for the nth term of an arithmetic sequence is given by:

$$a_n = a_1 + (n-1) imes d$$

Where:

$a_n$ is the nth term,
$a_1$ is the first term,
$d$ is the common difference,
$n$ is the position of the term in the sequence.

Rearrange the formula to solve for the first term ($a_1$)

We need to isolate $a_1$ on one side of the equation. This can be done by rearranging the formula:

$$a_1 = a_n - (n-1) imes d$$

Substitute the known values into the rearranged formula

Insert the known term value ($a_n$), its position ($n$), and the common difference ($d$) into the formula to find $a_1$.

Calculate the value of the first term ($a_1$)

Perform the arithmetic operations to determine the value of $a_1$.

To find the first term of an arithmetic sequence, use the formula $a_1 = a_n - (n-1) imes d$

More Information

Arithmetic sequences are widely used in various real-life contexts, such as determining payments for installment purchases, arranging seats in a theater, and figuring out scheduling patterns.

Tips

A common mistake is to substitute the wrong values for $n$ or $d$. Always verify the given values to avoid errors in calculation.

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