Next term of the following sequence is : A^(3/2) B_(2), C^(9/2) F_(18), E^(15/2) J_(50), G^(21/2) N_(98), ?

Correct — option (3), I^(27/2) R_(162).

The booklet prints each first letter with a superscript and each second letter with a subscript; here they appear as ^( ) and _( ). Each term has four parts, and each part gets its own rule.

Step 1 — first letters.
A, C, E, G = positions 1, 3, 5, 7, rising by 2
Next: position 9 = I

Step 2 — second letters.
B, F, J, N = positions 2, 6, 10, 14, rising by 4
Next: position 18 = R

Step 3 — superscripts.
3/2, 9/2, 15/2, 21/2, each rising by 6/2 = 3
Next: 21/2 + 6/2 = 27/2

Step 4 — subscripts.
2, 18, 50, 98: differences 16, 32, 48, rising by 16
Next difference 64: 98 + 64 = 162

Next term: I^(27/2) R_(162)

Check with the first letter's position n = 1, 3, 5, 7, 9: second letter = 2n, superscript = 3n/2, subscript = 2n². For n = 9: R (18), 27/2 and 2 × 81 = 162.

The idea to remember: in a mixed letter–number series, split each term into its parts and find a rule for each part.

Why the others are wrong

• (1)
  H^(29/2) S_(172) — Option (1) misses all four parts. H is position 8, but the first letters step by 2 from G (7) to I (9). S is position 19, off the step of 4 that takes N (14) to R (18).

  The superscripts rise by 3, so 21/2 goes to 27/2, not 29/2. The subscript differences 16, 32, 48 continue with 64, giving 162, not 172.

• (2)
  H^(27/2) R_(182) — Option (2) has the right second letter, R, and the right superscript, 27/2, but breaks the other two parts.

  The first letters rise by 2 from G (7) to I (9), not H (8). The subscripts 2, 18, 50, 98 are 2 × 1², 2 × 3², 2 × 5², 2 × 7², so the next is 2 × 9² = 162, not 182.

• (4)
  I^(29/2) R_(172) — Option (4) has both letters right, I and R, but both numbers are off.

  The superscripts rise by exactly 3 each time: 3/2, 9/2, 15/2, 21/2, 27/2; reaching 29/2 would need a step of 4. The subscript differences 16, 32, 48, 64 give 98 + 64 = 162, not 172.

A letter series works on alphabet positions, A = 1 to Z = 26. Steps can go forward or backward, stay constant or grow.

A number series can rise by a fixed difference, by differences that themselves follow a pattern, or by a formula such as n² or n³.

When each term has several parts, every part can follow its own rule. The parts can also share one counter: here all four depend on the first letter's position, 1, 3, 5, 7, 9.

Key facts

• Alphabet positions: A = 1, E = 5, J = 10, O = 15, T = 20, Y = 25, Z = 26.
• First letters A, C, E, G, I rise by 2; second letters B, F, J, N, R rise by 4.
• Superscripts 3/2, 9/2, 15/2, 21/2, 27/2 rise by 3 each time.
• Subscripts 2, 18, 50, 98, 162 equal 2 × 1², 2 × 3², 2 × 5², 2 × 7², 2 × 9².
• Constant second differences in a number sequence point to a quadratic rule; here the second difference is 16.

Common traps

• Stepping the first letter by 1 to H. The first letters skip a letter each time, A, C, E, G, so the next is I.
• Stepping the superscript by 4. The numerators 3, 9, 15, 21 rise by 6, so each superscript rises by 3 and the next is 27/2; a step of 4 gives the 29/2 in options (1) and (4).
• Treating the subscript difference as constant. The differences 16, 32, 48 themselves rise by 16, so the next is 64; repeating 48 would give 146.

• practice — not a real PYQ

  Next term of the following sequence is : 1A1, 3C9, 5E25, 7G49, ?

  • (a)9I81
  • (b)9H81
  • (c)8I64
  • (d)9I72
  Answer(1) — The first number rises by 2 (1, 3, 5, 7, 9); the letter sits at that position (A = 1, C = 3, E = 5, G = 7, I = 9); the last number is the square of the first (81).

  Option (2) puts H (8) at position 9, option (3) steps the first number by 1, and option (4) gives 72, which is not 9².

• practice — not a real PYQ

  Next term of the following sequence is : AZ, CX, EV, GT, ?

  • (a)HS
  • (b)IR
  • (c)IS
  • (d)JQ
  Answer(2) — The first letters move forward by 2 (A, C, E, G, I) and the second letters move back by 2 (Z, X, V, T, R). Option (1) moves each letter by 1, option (3) moves the second letter back by only 1, and option (4) moves both letters by 3.