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?

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 =