How to find a1 in an arithmetic sequence?
Understand the Problem
The question is asking for the method to find the first term (a1) in an arithmetic sequence. This involves understanding the characteristics of an arithmetic sequence, where each term after the first is derived by adding a constant difference to the previous term.
Answer
The first term \( a_1 \) is $2$.
Answer for screen readers
The first term ( a_1 ) is given by the equation: $$ a_1 = 2 $$
Steps to Solve
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Identify the variables Identify the values provided in the problem. In an arithmetic sequence, you typically need the first term ( a_1 ), a common difference ( d ), and the position of another term (or terms).
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Use the formula for the nth term The formula for the nth term of an arithmetic sequence is given by: $$ a_n = a_1 + (n-1)d $$ Here, ( a_n ) represents the nth term, ( a_1 ) is the first term, ( d ) is the common difference, and ( n ) is the term's position.
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Rearrange the formula to solve for the first term To find the first term ( a_1 ), rearrange the formula: $$ a_1 = a_n - (n-1)d $$
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Substitute known values Now, substitute the known values of ( a_n ), ( d ), and ( n ) into the rearranged formula:
- If ( a_n = 10 ), ( d = 2 ), and ( n = 5 ), then: $$ a_1 = 10 - (5-1) \times 2 $$
- Calculate the result Finally, perform the calculation to find ( a_1 ): $$ a_1 = 10 - 4 \times 2 $$ $$ a_1 = 10 - 8 $$ $$ a_1 = 2 $$
The first term ( a_1 ) is given by the equation: $$ a_1 = 2 $$
More Information
In an arithmetic sequence, each term increases (or decreases) by a fixed amount called the common difference. Knowing any term, the common difference, and the term's position allows you to easily calculate the first term.
Tips
- Forgetting to correctly identify ( n ) or ( d ). Ensure you know the position and common difference before performing calculations.
- Misapplying the formula. Always rearrange correctly to solve for ( a_1 ).
- Incorrect calculations when substituting values. Always double-check arithmetic.
