Arithmetic Progressions · Class 10 Mathematics: 60 flashcards
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1. What is an arithmetic progression?
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A list in which each term after the first is the previous term plus a fixed number.
2. What is the common difference?
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The fixed number added each time, written d.
3. How do you compute d?
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Any term minus the one before it: $d = a_{k+1} - a_k$.
4. Write the general form of an AP.
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$a,\ a + d,\ a + 2d,\ a + 3d,\ \ldots$
5. In the general form, what multiplies d in the fourth term?
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3 — one less than the position.
6. How do you test whether a list is an AP?
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Compute EVERY consecutive difference; they must all be equal.
7. Is $3, 7, 11, 14$ an AP?
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No — the differences are 4, 4 and 3.
8. Is $5, 5, 5, 5$ an AP?
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Yes, with $d = 0$.
9. Is $2, 4, 8, 16$ an AP?
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No — the differences are 2, 4, 8. Each term is multiplied, not increased by a fixed amount.
10. Is $1, 4, 9, 16$ an AP?
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No — the differences are 3, 5, 7.
11. Find d for $10, 7, 4, 1$.
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$d = 7 - 10 = -3$.
12. Write the first four terms when $a = -5$, $d = 4$.
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$-5, -1, 3, 7$.
13. Write the first four terms when $a = 10$, $d = -3$.
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$10, 7, 4, 1$.
14. Can d be a fraction?
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Yes — $\frac12, 1, \frac32, 2$ is an AP with $d = \frac12$.
15. What is the difference between a finite and an infinite AP?
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A finite AP has a last term; an infinite one does not and is written with three dots.
16. If a, b, c are in AP, what is b?
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$b = \frac{a + c}{2}$ — the arithmetic mean of a and c.
17. For $7, x, 19$ in AP, find x.
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$x = \frac{7 + 19}{2} = 13$.
18. A taxi charges ₹15 then ₹8 per extra km. Write the AP of fares.
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$15, 23, 31, 39, \ldots$ with $a = 15$, $d = 8$.
19. A pump removes a quarter of the air each stroke. Is the amount left an AP?
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No — $1, \frac34, \frac9{16}, \frac{27}{64}$ has shrinking differences.
20. What must a full 'is this an AP?' answer contain?
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Every difference shown, the conclusion stated, and a and d given if it is one.
21. State the formula for the nth term of an AP.
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$a_n = a + (n - 1)d$.
22. Why is the multiplier $n - 1$ rather than n?
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The first term costs no steps; reaching the nth takes $n-1$ steps of size d.
23. What does $a + nd$ give you?
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The term AFTER the one you wanted.
24. Find the 100th term of $3, 7, 11, \ldots$
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$3 + 99(4) = 399$.
25. Find the 20th term of $15, 12, 9, \ldots$
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$15 + 19(-3) = -42$.
26. How do you test whether x is a term of an AP?
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Solve $a + (n-1)d = x$; x is a term only if n is a positive integer.
27. Is 39 a term of $3, 7, 11, \ldots$?
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Yes — $n = 10$, so it is the 10th term.
28. Is 40 a term of $3, 7, 11, \ldots$?
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No — $n = 10.25$ is not a whole number.
29. How do you find the number of terms in a finite AP?
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Put the last term into the formula and solve for n.
30. How many terms are in $10, 17, 24, \ldots, 206$?
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29 — from $(n-1)7 = 196$.
31. Why is dividing $\frac{l - a}{d}$ not the number of terms?
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It counts the gaps; there is always one more term than gaps.
32. Two terms of an AP are given. What is the quickest method?
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Write both with the formula and subtract — the a cancels.
33. The 10th term is 41 and the 18th is 73. Find d.
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$8d = 32$, so $d = 4$.
34. The 10th term is 41 and the 18th is 73. Find a.
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$a + 9(4) = 41$, so $a = 5$.
35. Write the gap between the kth and mth terms.
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$a_k - a_m = (k - m)d$.
36. By how much does the 18th term exceed the 10th?
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$8d$, in any AP.
37. Give the formula for the nth term from the END.
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$l - (n - 1)d$, where l is the last term.
38. Find the 4th term from the end of $5, 9, 13, \ldots, 81$.
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$81 - 3(4) = 69$.
39. 'Which term is 39?' — what kind of answer is wanted?
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A position: the 10th term.
40. What should you write before substituting into the formula?
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The values of a and d, and the formula itself.
41. State the sum formula when a and d are known.
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$S_n = \frac{n}{2}[2a + (n-1)d]$.
42. State the sum formula when the last term is known.
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$S_n = \frac{n}{2}(a + l)$.
43. How are the two sum formulas related?
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Substituting $l = a + (n-1)d$ into the short form gives the long form.
44. Why does pairing terms work?
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Moving inwards from the left adds d while moving inwards from the right subtracts d, so every pair has the same total.
45. What is $1 + 2 + \ldots + n$?
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$\frac{n(n+1)}{2}$.
46. What is $1 + 2 + \ldots + 100$?
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$\frac{100 \times 101}{2} = 5050$.
47. Find the sum of the first 10 terms of $2, 5, 8, \ldots$
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$S_{10} = 5[4 + 27] = 155$.
48. Find the sum of $5, 9, 13, \ldots, 81$.
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There are 20 terms, so $S_{20} = 10(5 + 81) = 860$.
49. What kind of equation results when $S_n$ is given and n is unknown?
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A quadratic in n, because n appears twice in the formula.
50. Which roots must be rejected when solving for n?
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Any that are not positive whole numbers.
51. How many terms of $9, 17, 25, \ldots$ sum to 636?
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12 — from $4n^2 + 5n - 636 = 0$.
52. How many terms of $24, 21, 18, \ldots$ sum to 78?
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Both 4 and 13 — both roots are positive integers.
53. Why can two values of n give the same sum?
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When d is negative the later terms are negative and pull the total back down.
54. How do you recover $a_n$ from the sums?
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$a_n = S_n - S_{n-1}$.
55. What is $a_1$ in terms of the sums?
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$a_1 = S_1$.
56. If $S_n = 3n^2 + 5n$, find $a_{10}$.
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$S_{10} - S_9 = 350 - 288 = 62$.
57. What is the difference between $S_{30}$ and $a_{30}$?
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$S_{30}$ is the total of thirty terms; $a_{30}$ is the thirtieth term alone.
58. Why keep $\frac{n}{2}$ outside the bracket?
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Dividing early turns whole numbers into fractions and invites slips.
59. Penalties of ₹200, ₹250, ₹300 … for 30 days total what?
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$S_{30} = 15[400 + 1450] = ₹27{,}750$.
60. What must an answer solving for n contain?
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The quadratic, both roots, and a sentence accepting or rejecting each.
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