Final answer:
To find the n-th term of the sequence 9, 18, 27,..., use the formula a_n = 9 + (n-1)*9. To calculate the sum of all three-digit numbers in the sequence, identify the first and last three-digit terms and apply the sum formula for an arithmetic sequence, resulting in a total sum of 55449.
Step-by-step explanation:
Finding the n-th Term and The Sum of Three-Digit Numbers in a Sequence
The given sequence is 9, 18, 27,... which is an arithmetic sequence where each term increases by 9.
To find the n-th term of this sequence, we use the formula for the n-th term of an arithmetic sequence, which is an = a1 + (n - 1)d, where a1 is the first term and d is the common difference.
In this case, a1 = 9 and d = 9, so the n-th term is an = 9 + (n - 1) * 9.
To find the sum of all the three-digit numbers in this sequence, we need to first determine the first three-digit number in the sequence, which is 100.
Since our sequence starts at 9 and increases by 9, the first three-digit number is the 11th term (100 = 9 + (n - 1)*9, solving for n gives n=11).
The last three-digit number in the sequence is 999, which is the 111th term (999 = 9 + (n - 1)*9, solving for n gives n=111). To find the sum of these terms, we can use the sum of an arithmetic sequence formula:
Sn = n/2 * (a1 + an).
Here we have 101 terms (from the 11th to the 111th term), the first term is 99 (the 11th term) and the last term is 999 (the 111th term).
Thus, the sum is S101 = 101/2 * (99 + 999), which simplifies to S101 = 50.5 * (1098), resulting in 55449 as the sum of all three-digit numbers in the sequence.