Arithmetic series and multiples of 5

Key concept: This solution uses the arithmetic series formula and the structure of multiples of 5. The main idea is to identify the first term, last term, number of terms, and then apply the sum formula carefully.

Step by step

1) Sum of the first 8000 terms

The work shown indicates an arithmetic-series setup:

$S_{8000}=\frac12\bigl(1000(4000)\bigr)$

The clearer form of the arithmetic series formula is

$S_n=\frac{n}{2}(a_1+a_n)$

Using the values shown in the working, the sum is

$\boxed{1\,602\,000}$

2) Multiples of 5 from 5 to 4000

The sequence is

$5,10,15,\ldots,4000$

The $n$th term is

$t_n=5+(n-1)5=5n$

Set $t_n=4000$:

$4000=5n$

$n=800$

So there are $800$ terms.

The sum is

$S_{800}=\frac{800}{2}(5+4000)$

$S_{800}=400\cdot 4005=1\,602\,000$

3) Difference shown in the working

The expression

$8\,002\,000-1\,602\,000=6\,400\,000$

is a subtraction step comparing two totals. The result is

$\boxed{6\,400\,000}$

Pitfall alert

A frequent mistake is writing the series formula correctly but substituting the wrong first and last terms. Another common error is finding the number of terms in a multiples-of-5 list by dividing by 4000 instead of using the nth-term equation $t_n=5n$.

Try different conditions

If the sequence started at a different multiple of 5, say 10, 15, 20, \ldots, the nth-term formula would change to $t_n=5n+5$ or an equivalent form, and the sum would need the new first term and last term. The same arithmetic-series method still applies.

Further reading

arithmetic series, nth term, sum formula