What is the total number of terms and their sum for the sequence beginning with 4, 7, 10 and ending with 64?

To determine the total number of terms and their sum in the sequence starting with 4, 7, 10, and ending with 64, we first identify the nature of the sequence. This is an arithmetic sequence where the first term is 4 and each subsequent term increases by a common difference, which can be found by subtracting the first term from the second:

7 - 4 = 3.

Therefore, the sequence has a common difference of 3. The nth term of an arithmetic sequence can be expressed with the formula:

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

where:

• ( a_n ) is the nth term,

• ( a_1 ) is the first term (4),

• ( d ) is the common difference (3), and

• ( n ) is the number of terms.

Setting ( a_n ) to 64, we solve for ( n ):

[ 64 = 4 + (n-1) \times 3 ]

Subtracting 4 from both sides gives:

[ 60 = (n-1) \times 3 ]

Dividing both sides by 3 results in:

[