Directions : There is a wrong number in these series. Find the wrong number &…

Concept

In a wrong-number series the terms follow ONE fixed rule; the single term that violates that rule is the wrong number, and the value the rule demands in its place is the 'correct term'. Here we first recover the rule of each series to get its correct term, then evaluate a quadratic to get R, and finally test each statement by substitution.

Series A — find P

The consecutive differences should themselves rise by a constant 7.5, i.e. the differences form an arithmetic progression.

1. Differences of 24, 31.5, 46.5, 69, …, 136.5, 181.5 should be 7.5, 15, 22.5, 30, 37.5, 45 (each 7.5 more than the last).

2. From 69 the next term must be 69 + 30 = 99, but the series shows 98 — so 98 is the wrong number and the correct term is P = 99.

Series B — find Q

This is a subtraction series whose successive subtractions shrink by dividing by 7, 6, 5, 4, 3.

1. The amounts subtracted are 5040, 720, 120, 24, 6, 2 (5040÷7 = 720, 720÷6 = 120, 120÷5 = 24, 24÷4 = 6, 6÷3 = 2).

2. So after 22 the term must be 22 − 6 = 16, but the series shows 18 — so 18 is the wrong number and the correct term is Q = 16.

Note: Series C (11, 18, 44, …) is part of the directions set but is not used in this question — only P (from Series A) and Q (from Series B) feed the statements.

Equation — find R

1. Right side: −92 + 52 = −81 + 25 = −56.

2. So x2 − 15x = −56, i.e. x2 − 15x + 56 = 0.

3. Factorising: (x − 7)(x − 8) = 0, giving roots 7 and 8; the larger root is 8.

4. R = square of the larger root = 82 = R = 64.

Test each statement (P = 99, Q = 16, R = 64)

• (i) Q + R = 16 + 64 = 80, which is not 99 = P, so (i) is incorrect.

• (ii) P/3 + 15 = 33 + 15 = 48, and R − Q = 64 − 16 = 48; both equal 48, so (ii) is correct.

• (iii) √R + P = 8 + 99 = 107, and Q × 7 − 5 = 112 − 5 = 107; both equal 107, so (iii) is correct.

Only (ii) and (iii) hold, so the answer is the choice naming statements (ii) & (iii).