In an arithmetic sequence where the first term is 6 and the seventh term is 24, what is the sum of the first seven terms?

In an arithmetic sequence where the first term is 6 and the seventh term is 24, what is the sum of the first seven terms?

• A. 85
• B. 90
• C. 105
• D. 110

Answer: (C)

To find the sum of the first seven terms in the arithmetic sequence, we first need to determine the common difference of the sequence. In an arithmetic sequence, the n-th term can be expressed using the formula: \[ a_n = a_1 + (n - 1) \cdot d \] where \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) is the term number. Given that the first term \( a_1 \) is 6 and the seventh term \( a_7 \) is 24, we can set up the equation for the seventh term: \[ a_7 = 6 + (7 - 1) \cdot d \] This simplifies to: \[ 24 = 6 + 6d \] Subtracting 6 from both sides gives: \[ 18 = 6d \] Dividing both sides by 6 leads to: \[ d = 3 \] Now that we have the common difference, we can determine the first seven terms of the sequence: - First term: \( a_1 = 6 \) - Second term: \( a_2 = 6 + 3 =

To find the sum of the first seven terms in the arithmetic sequence, we first need to determine the common difference of the sequence.

In an arithmetic sequence, the n-th term can be expressed using the formula:

[ a_n = a_1 + (n - 1) \cdot d ]

where ( a_1 ) is the first term, ( d ) is the common difference, and ( n ) is the term number.

Given that the first term ( a_1 ) is 6 and the seventh term ( a_7 ) is 24, we can set up the equation for the seventh term:

[ a_7 = 6 + (7 - 1) \cdot d ]

This simplifies to:

[ 24 = 6 + 6d ]

Subtracting 6 from both sides gives:

[ 18 = 6d ]

Dividing both sides by 6 leads to:

[ d = 3 ]

Now that we have the common difference, we can determine the first seven terms of the sequence:

• First term: ( a_1 = 6 )

• Second term: ( a_2 = 6 + 3 =