Section 15, easy algebra

Progressions (AP & GP)

Progression questions are mostly formula substitution: the nth term and sum of an AP, the sum and infinite sum of a GP. CAT dresses them up as savings plans, rows of seats, bouncing balls and “how many numbers between A and B”.

How CAT asks it

Terms or sums at two positions are given and you find others; you count or sum the numbers in a range with a divisibility property; you find the common terms of two APs; infinite GP sums (bouncing balls); word problems with steady increases; Sₙ given as a formula in n.

Concepts you must know

  • AP: a, a + d, a + 2d, … with nth term Tₙ = a + (n − 1)d.
  • AP sum = n × (first + last)/2, and the average of an AP = (first + last)/2.
  • Three terms in an AP: write them as a − d, a, a + d (they sum to 3a).
  • Number of terms from p to q with gap d = (q − p)/d + 1.
  • GP: a, ar, ar², … with Tₙ = arⁿ⁻¹ and Sₙ = a(rⁿ − 1)/(r − 1).
  • The common terms of two APs form an AP whose gap is the LCM of the two gaps.
  • If Sₙ = An² + Bn, the series is an AP with d = 2A and first term A + B; any term is Tₙ = Sₙ − Sₙ₋₁.
  • 1 + 2 + … + n = n(n + 1)/2; the sum of the first n squares is n(n + 1)(2n + 1)/6; of the first n cubes, [n(n + 1)/2]²; of the first n odd numbers, n².
  • The AP sum and the infinite GP sum:
Sn=n2[2a+(n−1)d]S_n = \frac{n}{2}\left[2a + (n-1)d\right]
S∞=a1−r(∣r∣<1)S_\infty = \frac{a}{1-r}\quad (|r| < 1)

Thought process

  1. Decide AP (constant difference) or GP (constant ratio).
  2. Write a and d (or r). Two given facts give two equations.
  3. For counting in a range, find the first and last valid terms, then count.
  4. For bouncing balls, count the first drop once and every rebound height twice (up and down).

Patterns CAT repeats

Numbered so your error log can point at them: “Pattern 2”.

  1. 1Two terms given: find d and a, then any term or sum.
  2. 2Counting or summing numbers in a range divisible by k.
  3. 3Common terms of two APs.
  4. 4Infinite GP: bouncing balls, repeating decimals.
  5. 5Steady increases in word problems: savings, rows of seats.
  6. 6Sₙ given as a formula in n.
  7. 7Average of AP terms (overlaps with Section 9).

Worked examples

Attempt each one before opening its solution, then compare methods.

  1. Example 1

    Find the sum of all two-digit multiples of 3.

    Show solution

    They run from 12 to 99: (99 − 12)/3 + 1 = 30 terms. Sum = 30 × (12 + 99)/2. Answer: 1,665.

  2. Example 2

    The 5th term of an AP is 19 and the 12th is 47. Find the sum of the first 20 terms.

    Show solution

    7d = 28, so d = 4 and a = 19 − 16 = 3. S₂₀ = 10 × (2 × 3 + 19 × 4) = 10 × 82. Answer: 820.

  3. Example 3

    How many terms are common to 4, 9, 14, …, 199 and 3, 10, 17, …, 220?

    Show solution

    The first common term is 24 (in the second list: 3, 10, 17, 24). Common terms step by LCM(5, 7) = 35: 24, 59, 94, 129, 164, 199. The next, 234, passes 199. Answer: 6.

  4. Example 4

    A ball dropped from 20 m rebounds to 3/4 of its previous height every time. What total distance does it travel before stopping?

    Show solution

    20 down, then each rebound height up and down: 20 + 2 × [15/(1 − 3/4)] = 20 + 2 × 60. Answer: 140 m.

  5. Example 5

    Riya saves ₹500 in the first month and ₹100 more each month after that. After how many months will her total savings reach ₹11,000?

    Show solution

    Sₙ = n/2 × [1,000 + 100(n − 1)] = 50n² + 450n = 11,000, so n² + 9n − 220 = 0 = (n + 20)(n − 11). Check: 11 × (500 + 1,500)/2 = 11,000. Answer: 11 months.

  6. Example 6

    The sum of the first n terms of an AP is 3n² + 5n. Find the 10th term.

    Show solution

    T₁₀ = S₁₀ − S₉ = 350 − 288. Check with d = 6 and a = 8: 8 + 9 × 6 = 62. Answer: 62.

Traps

  • Off-by-one errors when counting terms: it is (last − first)/d + 1.
  • Using the infinite GP formula when |r| ≥ 1.
  • Counting a bouncing ball’s first drop twice, or its rebounds once.

Practice set

Timed, about 2.5 minutes per question. Log every miss in the error log.

  1. 1

    How many three-digit numbers are divisible by 7?

    Show answer

    128: from 105 (7 × 15) to 994 (7 × 142).

  2. 2

    The 3rd term of an AP is 11 and the 7th is 27. Find the 15th term.

    Show answer

    59: d = 4 and a = 3.

  3. 3

    Find the sum of the infinite GP 12 + 6 + 3 + …

    Show answer

    24: 12 ÷ (1 − 1/2).

  4. 4

    Three numbers in an AP add to 24 and multiply to 440. Find them.

    Show answer

    5, 8 and 11: the middle term is 24/3 = 8, and 8(64 − d²) = 440 gives d = 3.

  5. 5

    A theatre has 20 seats in the first row and 2 more in each row behind. How many seats are there in 15 rows?

    Show answer

    510: the last row has 48 seats, and 15 × (20 + 48)/2 = 510.

  6. 6

    Find the sum of the first 20 odd numbers.

    Show answer

    400: the first n odd numbers add to n².

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