For a given arithmetic sequence, the first term, a1, is equal to -22, and the 93rd term, a93, is equal to -574. Find the value of the 34th term, a34.

Understand the Problem

The question is asking to find the 34th term of an arithmetic sequence given the first term and the 93rd term. We will need to first determine the common difference of the sequence and then use it to calculate the desired term.

Answer

The 34th term $ a_{34} $ is $-220$.

Steps to Solve

Identify the given values

The first term of the arithmetic sequence is given as ( a_1 = -22 ) and the 93rd term as ( a_{93} = -574 ).

Use the formula for the nth term of an arithmetic sequence

The formula for the nth term ( a_n ) of an arithmetic sequence is given by: $$ a_n = a_1 + (n - 1) \cdot d $$ where ( d ) is the common difference.

Set up the equation for the 93rd term

For the 93rd term, we can substitute ( n = 93 ) into the formula. Thus: $$ a_{93} = a_1 + (93 - 1) \cdot d $$ Substituting in the known values: $$ -574 = -22 + 92d $$

Solve for the common difference ( d )

Rearranging the equation to isolate ( d ): $$ -574 + 22 = 92d $$ This simplifies to: $$ -552 = 92d $$ Now, dividing both sides by 92 gives: $$ d = \frac{-552}{92} = -6 $$

Use the common difference to find the 34th term

Now, substitute ( n = 34 ) into the original formula: $$ a_{34} = a_1 + (34 - 1) \cdot d $$ Substituting ( a_1 ) and ( d ): $$ a_{34} = -22 + (33) \cdot (-6) $$ Calculating the expression: $$ a_{34} = -22 - 198 = -220 $$

The value of the 34th term ( a_{34} ) is $-220$.

More Information

An arithmetic sequence is characterized by a constant difference between consecutive terms. Knowing the two terms allows us to easily find the common difference and subsequently any term in the sequence.

Tips

Forgetting to convert ( n - 1 ) properly when substituting to find the common difference.
Mixing up the signs when calculating the common difference or the desired term.

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