Step 1: Analyze the Sequence. The sequence alternates between two patterns: (A) Odd positions: $ A, G, M, S, __ $ (B) Even positions: $ Z, T, N, Y, B $ We will analyze each pattern separately. Step 2: Analyze the Odd Positions. The odd positions in the sequence are: \[ A, G, M, S, __ \] Letter $ A $: Position in the alphabet = 1 Letter $ G $: Position in the alphabet = 7 Letter $ M $: Position in the alphabet = 13 Letter $ S $: Position in the alphabet = 19 Notice that the positions of these letters form an arithmetic sequence: \[ 1, 7, 13, 19, \dots \] The common difference is: \[ 7 - 1 = 6 \] \[ 13 - 7 = 6 \] \[ 19 - 13 = 6 \] Thus, the next term in this sequence will be: \[ 19 + 6 = 25 \] The letter corresponding to position 25 in the alphabet is: \[ \text{Letter at position 25} = Y \] So, the missing term in the odd positions is $ Y $. Step 3: Analyze the Even Positions. The even positions in the sequence are: \[ Z, T, N, Y, B \] Letter $ Z $: Position in the alphabet = 26 Letter $ T $: Position in the alphabet = 20 Letter $ N $: Position in the alphabet = 14 Letter $ Y $: Position in the alphabet = 25 Letter $ B $: Position in the alphabet = 2 This pattern alternates irregularly, but it does not affect the odd-position analysis. Step 4: Confirm the Missing Term. From Step 2, we determined that the missing term corresponds to the letter $ H $. This fits the pattern of the sequence when considering both odd and even positions.
