What is the total distance the competitor has to run ?

Q: Can this be solved without the AP sum formula?

Yes, by manually adding all ten round-trip distances ($10+16+22+\ldots+64$), but using $S_n = \frac{n}{2}[2a+(n-1)d]$ is faster and less error-prone.

Arithmetic Progressions

Arithmetic Progressions - Sum of n terms (Potato Race)

Official Answer

The round-trip distances $10, 16, 22, \ldots$ form an AP with $a = 10\text{ m}$ and $d = 6\text{ m}$ for n = 10 potatoes. Applying $S_n = \frac{n}{2}[2a + (n - 1)d]$ , $S_{10} = 5[20 + 9(6)] = 5 \times 74 = 370\text{ m}$ . Hence the competitor runs a total distance of 370 m.

sum of AParithmetic progressiontotal distancepotato raceSn formula10 potatoes

Marking Scheme

11 mark: correctly identifying $a = 10, d = 6, n = 10$ and writing the sum formula $S_n = \frac{n}{2}[2a+(n-1)d]$ .
21 mark: correct substitution and final answer of 370 m.

Hint

Use $S_n = \frac{n}{2}[2a + (n-1)d]$ with $a = 10, d = 6, n = 10$ .

Quick Oral Answer

The round-trip distances form an AP with $a = 10\text{ m}, d = 6\text{ m}, n = 10$ ; using $S_n = \frac{n}{2}[2a+(n-1)d]$ , the total distance is 370 m.

Analysis & Explanation

The culminating step of the case study: modelling the ten round-trip distances as an AP and summing them.

Concept

Round-trip distances $10, 16, 22, \ldots$ form an AP with $a = 10\text{ m}$ (Q43) and $d = 6\text{ m}$ (from Q43 to Q44).
Total distance for 10 potatoes = Sₙ using the standard AP sum formula.

Key Points

$S_n = \frac{n}{2}[2a + (n - 1)d]$ , with $n = 10, a = 10, d = 6$ .
$S_{10} = 5[20 + 54] = 370\text{ m}$ .

Common Mistakes

Using $(n - 1) = 10$ instead of 9 inside the formula.
Taking $d = 3\text{ m}$ (the one-way gap) instead of the correct round-trip value $d = 6\text{ m}$ .

Real-World

Shows how AP models practical sports-event logistics like relay/potato races.

Common Mistakes

1Using $d = 3\text{ m}$ (the one-way gap) instead of $d = 6\text{ m}$ (the round-trip increase).
2Arithmetic slip in $(n-1)d$ , e.g. using n instead of (n − 1).
3Manually adding ten terms and making an addition error instead of using the sum formula.

Interesting Facts

This exact potato-race scenario (bucket 5 m away, potatoes 3 m apart, 10 potatoes) is one of the most quoted real-life applications of Arithmetic Progressions in NCERT Class 10 Mathematics.

The total distance of 370 m is roughly equivalent to running almost one full lap of a standard 400 m athletics track.

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Frequently Asked Questions

Why is the common difference 6 m and not 3 m?

Because the 3 m gap between consecutive potatoes affects both the outward and return trip, doubling to a 6 m increase in round-trip distance for each successive potato.

Can this be solved without the AP sum formula?

Yes, by manually adding all ten round-trip distances ( $10+16+22+\ldots+64$ ), but using $S_n = \frac{n}{2}[2a+(n-1)d]$ is faster and less error-prone.