Arithmetic Series
Find d, then formula or Gauss-pair your way to the answer.
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: Find d, then formula or Gauss-pair your way to the answer.
★ · Spot the Step
A list of numbers begins 2, 5, 8, 11, ... If the same rule keeps going, what is the next number, and how big is the step from one number to the next?
L0 · Constant Step
Maya walks across a pond on 8 stepping stones. Each stone is heavier than the one before by the same amount. The first stone is 5 kilograms, the second is 8 kilograms, and the third is 11 kilograms. How heavy is the 8th stepping stone?
L1 · Far-Term Formula
Aiden walks down a street. The house numbers form a pattern: 7, 13, 19, 25, 31, and so on. What number is on the 20th house?
L2 · Gauss Pairing
Maya stacks blocks in a 12-row staircase tower. Row 1 has 4 blocks, and each row above has 3 more blocks than the one below. How many blocks are in the whole tower?
L3 · Count by Step
A pet shelter assigns ID tags in multiples of four. The numbers run from 100 up to 196. How many ID tags does the shelter give out in total?
L4 · Sum a Slice
Carlos reads chapters every weekend. Weekend 1 he reads 5 chapters, weekend 2 he reads 8, weekend 3 he reads 11, and the pattern keeps adding 3 chapters each weekend. How many chapters does he read across weekends 5 through 12?
L5 · Subtract Two Series
Two choirs share a 60-minute concert. Choir A sings on the odd minutes (1, 3, 5, ...) and Choir B sings on the even minutes (2, 4, 6, ...). On each minute, the singing choir claps that many times. How many more total claps does Choir B do than Choir A?
2 Learn — watch the solutions
Gave each one a real try? Now watch the trick. (Stuck is fine — that's the point.)
★ · Spot the Step
Peek the trick — Same-gap check
A list is arithmetic only when every consecutive gap is the same number. Subtract neighbours once or twice to see whether the step is constant.
L0 · Constant Step
Peek the trick — Jump using a1 + (n-1)·d
Once the step d is fixed, any term equals the first term plus d added (n-1) times. You do not have to list every term to reach a near position.
L1 · Far-Term Formula
Peek the trick — Skip the list with a_n = a_1 + (n-1)·d
When the position is too big to write out, plug the start, the step, and the position into one formula. No enumeration needed.
L2 · Gauss Pairing
Peek the trick — Sum = n · (first + last) ÷ 2
Pair the first term with the last, the second with the second-to-last, and so on; every pair has the same total. Multiply that pair-total by the number of pairs.
L3 · Count by Step
Peek the trick — n = (end − start) ÷ step + 1
The fence-post +1 catches both endpoints. Divide the spread by the step first, then add one for the starting term.
L4 · Sum a Slice
Peek the trick — Slice sum = (m − k + 1) · (a_k + a_m) ÷ 2
When you only want the kth through mth terms, build the first and last of that slice with the far-term formula, then Gauss-pair inside the slice.
L5 · Subtract Two Series
Peek the trick — Compute S_A and S_B with Gauss, then subtract
When both totals are themselves arithmetic sums, find each one with Gauss pairing and subtract. Never add term by term.
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
