Write a general form of an explicit function for the $n$th term in terms of $a$ and $d$. Use the form below to write your function.
$f(n) =$
Answer (2)
The problem asks for an explicit formula for the n th term of an arithmetic sequence.
The general form of an arithmetic sequence is a n = a + ( n − 1 ) d .
Express the formula as a function: f ( n ) = a + ( n − 1 ) d .
The explicit function for the n th term is f ( n ) = a + ( n − 1 ) d .
Explanation
Understanding the Problem We are asked to find a general explicit formula for the n th term of an arithmetic sequence. The formula should be in terms of a and d , where a is the first term and d is the common difference. We need to express this formula as f ( n ) = ...
Recalling the Arithmetic Sequence Formula The general form of an arithmetic sequence is given by: a n = a + ( n − 1 ) d where: a n is the n th term, a is the first term, n is the term number, d is the common difference.
Expressing as a Function We need to express this formula as a function f ( n ) . So, we replace a n with f ( n ) :
f ( n ) = a + ( n − 1 ) d
Final Answer We can simplify the expression further: f ( n ) = a + n d − d f ( n ) = n d + ( a − d ) So, the explicit function for the n th term of the arithmetic sequence is: f ( n ) = a + ( n − 1 ) d
Examples
Arithmetic sequences are useful in various real-life scenarios. For example, consider a savings plan where you deposit a fixed amount each month. If you start with an initial deposit of a and add d every month, the total amount saved after n months can be modeled using the arithmetic sequence formula f ( n ) = a + ( n − 1 ) d . This helps you predict your savings over time.
The explicit function for the n th term of an arithmetic sequence is given by the formula f ( n ) = a + ( n − 1 ) d , where a is the first term and d is the common difference. This formula allows us to calculate any term in the sequence based on its position. Understanding this formula is essential for solving problems related to arithmetic sequences effectively.
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