Two ways to find it, the "thinking" way, and the "robotic" way:
Method 1. By writing the series in reverse order and adding, a method
in which you understand why it works.
Method 2. By using two memorized formulas that you have no idea
why they work.
Method 1:
8 + 9 + 10 + 11 + ... 397 + 398 + 399 + 400
8 is term #1, and 1 is 7 less than 8
9 is term #2, and 2 is 7 less than 9
10 is term #3, and 3 is 7 less than 10
11 is term #4, and 4 is 7 less than 11
Therefore we conclude that 400 is term #393
because 393 is 7 less than 400.
So there are 393 terms.
Write the sequence and underneath write the sequence in
reverse order:
Sum = 8 + 9 + 10 + 11 + ... + 397 + 398 + 399 + 400
Sum = 400 + 399 + 398 + 397 + ... + 11 + 10 + 9 + 8
Now draw a line underneath and add them term by term:
Sum = 8 + 9 + 10 + 11 + ... + 397 + 398 + 399 + 400
Sum = 400 + 399 + 398 + 397 + ... + 11 + 10 + 9 + 8
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2×Sum = 408 + 408 + 408 + 408 + ... + 408 + 408 + 408 + 408
So we have 393 terms each of which is 408.
2×Sum = 393×408 = 160344
Therefore the sum is half of that
Sum = ×156264 = 80172
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Method 2:
Memorize these two formula which you have no idea why they works.
an = a1 + (n-1)d
Sn = (a1 + an)
where a1 = first term = 8
n = the number of terms
d = the difference between any term and the one before it = 1
an = the nth term
Sn = the sum of the first n terms.
We use the first memorized formula to find n:
an = a1 + (n-1)d
400 = 8 + (n-1)(1)
400 = 8 + n - 1
400 = 7 + n
393 = n
Then we substitute in the sum formula we have memorized:
Sn = (a1 + an)
S393 = (8 + 400)
S393 = (408)
S393 = 80172
Edwin
