Let's solve the problem step-by-step:
### Step 1: Understand the Sequence
We are given a sequence where each number increases by 50. The initial number in the sequence is 70. The sequence is generated as follows:
[tex]$
70 \xrightarrow{+50} 120 \xrightarrow{+50} 170 \xrightarrow{+50} \ldots
$[/tex]
### Step 2: Write the General Formula for the nth Term
We can write the general formula for the nth term of this arithmetic sequence. The nth term, which we'll denote as [tex]\( a_n \)[/tex], can be expressed as:
[tex]$
a_n = a_1 + (n-1) \cdot d
$[/tex]
where:
- [tex]\( a_1 = 70 \)[/tex] (the first term)
- [tex]\( d = 50 \)[/tex] (the common difference)
- [tex]\( n \)[/tex] is the term number.
### Step 3: Find the Minimum Term Greater Than 1000
We need to find the smallest term [tex]\( a_n \)[/tex] that is greater than 1000. Thus, we need to solve the inequality:
[tex]$
a_n > 1000
$[/tex]
Substitute the formula for [tex]\( a_n \)[/tex]:
[tex]$
70 + (n-1) \cdot 50 > 1000
$[/tex]
### Step 4: Solve the Inequality
First, let's isolate the term involving [tex]\( n \)[/tex]:
[tex]$
70 + 50(n-1) > 1000
$[/tex]
Subtract 70 from both sides:
[tex]$
50(n-1) > 930
$[/tex]
Divide both sides by 50:
[tex]$
n-1 > 18.6
$[/tex]
Add 1 to both sides to solve for [tex]\( n \)[/tex]:
[tex]$
n > 19.6
$[/tex]
Since [tex]\( n \)[/tex] must be a whole number, we take the ceiling (next whole number), which means:
[tex]$
n = 20
$[/tex]
### Step 5: Verify the Answer
Let's verify by calculating the 20th term of the sequence:
Using the formula:
[tex]$
a_{20} = 70 + (20-1) \cdot 50
$[/tex]
Calculate the term:
[tex]$
a_{20} = 70 + 19 \cdot 50
$[/tex]
[tex]$
a_{20} = 70 + 950
$[/tex]
[tex]$
a_{20} = 1020
$[/tex]
### Conclusion
The first number in the sequence that is greater than 1000 is 1020, and it is the 20th term in the sequence.
So, the answer is:
- Term number [tex]\( n = 20 \)[/tex]
- First number greater than 1000 in the sequence [tex]\( = 1020 \)[/tex]
