Sequences & Series (CIMC L0–L5 + Gauss Foundation)
Find the step. Write the sum a second way. Watch the messy middle disappear.
1 Explore — try these first
Try before you watch. Pick a level below and give the problem an honest try on paper first — wrong turns and all. Then open the video to see the trick. Every level rides one habit: When a list grows by a steady rule, find the step and write the sum a second way — pair it, factor it, or subtract the overlap.
★ · Gauss Pairing
Add every whole number from 1 to 100. That is, find 1 + 2 + 3 + … + 100.
L0 · Find the Step
A teacher fills snack bags each day this week. Monday she fills 14 bags, Tuesday 19, Wednesday 24, Thursday 29. If the pattern continues, how many bags will she fill on Friday?
L1 · Count the Steps, Not the Calendar
A class makes welcome cards on the odd-numbered days only. On Day 1 they make 20 cards, on Day 3 they make 25, on Day 5 they make 30. If the pattern continues, how many cards will they make on Day 17?
L2 · Gauss Pairing (applied)
A community book corner has 15 shelves. The first shelf holds 7 books, the second 8, the third 9, and so on — each shelf holds one more book than the one before. What is the total number of books across all 15 shelves?
L3 · Factor Out, Then Pair
At a weekend meal program, the number of lunch boxes packed grows by 6 each day. On Day 1 she packs 12 boxes. After several days the total number of boxes packed is 210. How many days did the packing continue?
L4 · Write-It-Twice, Then Combine
A support line starts a phone tree to recruit volunteers. In Round 1, five new volunteers join. In each round after that, the number of new volunteers triples (×3). The tree stops before any round would add more than 1000 new volunteers. How many new volunteers join altogether?
L5 · Inclusion–Exclusion
Find the sum of every positive integer less than 200 that is a multiple of 11 OR a multiple of 13.
2 Learn — watch the solutions
Gave each one a real try? Now watch the trick. (Stuck is fine — that's the point.)
★ · Mental Math Challenge — Gauss Sum 1 to 100
Peek the trick — Pair the outsides and equalise
Write the sum forwards and backwards on two lines, then look at the columns — every pair adds to the same constant, and the count of columns is your multiplier.
L0 · The Snack Bag Pattern — Spot the Next Number
Peek the trick — Subtract neighbours, then extend
Subtract one term from the next to read the constant step, then add that step on to reach the next day, the next week, or any term you want.
L1 · The Welcome Card Pattern — Find the 17th Day
Peek the trick — Map calendar to step index
When the calendar skips, the number of steps from the first listed day to a target day is the gap in calendar days divided by the gap between listed days — count those steps, not the calendar.
L2 · The Bookshelf Total — Adding 15 Numbers Fast
Peek the trick — Forwards plus backwards, then halve
Write the sum forwards and backwards. Each column adds to (first + last), and there are n columns — so the total is n times (first + last), halved.
L3 · The Lunchbox Count — How Many Days?
Peek the trick — Scale down, Gauss-pair, scale back
When every term shares a common factor, divide both the pattern AND the total by that factor — the big numbers shrink to a clean small sum you can pair in your head, then scale back up.
L4 · The Phone Tree Total — When Things Triple
Peek the trick — Multiply by r, then subtract
For a geometric sum (each term times the same ratio), write the sum once and then multiply the whole sum by the ratio — line them up, subtract, and the messy middle cancels.
L5 · When Sets Overlap — Multiples of 11 OR 13
Peek the trick — Add the two sets, subtract the overlap
For a union of two patterns, sum each pattern separately, then subtract the sum of the overlap — multiples of the LCM — because the overlap is otherwise counted twice.
3 Master — practice on your own
Print the practice sheet and solve without the videos. Check your answers at the back — if one is wrong, the answer key names the trick so you know exactly which video to rewatch.
Download fresh practice problems PDF
