One walker was sponsored

$100 plus $5 for the first kilometre, $10
for the second kilometre, $15 for the
third kilometre, and so on. How far
would this walker need to
walk to earn $150?

(I know it is 4 km, but I can't figure out how to write the general term.)

You have his earnings as

100+5+10+15+...
Since he gets $100 just for entering, all he needs to do is walk enough miles to get another $50

5+10+15+... is an arithmetic series of n terms, where

Sn = n/2 (2*5+(n-1)5) >= 50
n >= 4

Check: 5+10+15+20 = 50

Thank you!

To find the distance the walker would need to walk to earn $150, we need to determine the pattern and find the general term for the sponsorship amounts.

The sponsorship amounts follow an arithmetic sequence, where each term is found by adding a constant difference to the previous term. The initial amount is $100, and for each kilometer, the sponsorship amount increases by $5.

So, the sponsorship amounts can be written as follows:
First term: $100
Second term: $100 + $5
Third term: $100 + $5 + $10
Fourth term: $100 + $5 + $10 + $15
And so on...

To find the general term of this arithmetic sequence, we need to find the formula that relates the term number (n) and the sponsorship amount (Sn).

We observe that for each term, the amount increases by $5 more than the previous term. So, the difference between terms (d) is also increasing by $5 for each additional term.

The general term (Sn) of an arithmetic sequence can be calculated using the formula:
Sn = a + (n - 1) * d

Where:
Sn is the sponsorship amount for the nth term,
a is the first term of the sequence,
d is the common difference between terms, and
n is the term number.

In this case, a = $100 and d = $5. So, the general term (Sn) can be written as:
Sn = 100 + (n - 1) * 5

Now, we can use this formula to find the term number (n) when the sponsorship amount (Sn) is $150:
150 = 100 + (n - 1) * 5

Let's solve this equation for n:
150 - 100 = (n - 1) * 5
50 = (n - 1) * 5
10 = n - 1

Adding 1 to both sides:
10 + 1 = n - 1 + 1
11 = n

Therefore, the walker would need to walk 11 kilometers to earn $150.