Directions: Identify the rule used in the number sequence.

What value belongs in place of the question mark?

4, 10, 36, ?, 960, 6480, 50400

The correct answer is 168.

To find the missing number in the sequence, let us analyze the pattern connecting consecutive terms.

The given sequence is:

4, 10, 36, ?, 960, 6480, 50400

Step 1: Identify the Pattern / Rule

Each term in the sequence is obtained by multiplying the previous term by an increasing multiplier integer, and then adding the factorial of that multiplier integer.

If we let $T_n$ denote the n-th term of the sequence, the rule for finding the next term $T_{n+1}$ using multiplier k is given by:

$T_{n+1}=(T_n\times k)+k!$

where the multiplier k increases by 1 for each step (starting from k = 2).

Step 2: Verify the Given Initial Terms

1. Transition from the 1st term (4) to the 2nd term (10) using k = 2:

$(4\times 2)+2!=8+2=10$

2. Transition from the 2nd term (10) to the 3rd term (36) using k = 3:

$(10\times 3)+3!=30+6=36$

Step 3: Calculate the Missing 4th Term (?)

Following the pattern, the next multiplier is k = 4. We multiply the 3rd term (36) by 4 and add 4!

Note that 4! = 4 × 3 × 2 × 1 = 24.

$(36\times 4)+4!=144+24=168$

Step 4: Verify the Remaining Terms of the Sequence

To ensure the pattern holds completely, let us calculate the subsequent terms using 168:

1. Transition to the 5th term using k = 5 (where 5! = 120):

$(168\times 5)+5!=840+120=960$

2. Transition to the 6th term using k = 6 (where 6! = 720):

$(960\times 6)+6!=5760+720=6480$

3. Transition to the 7th term using k = 7 (where 7! = 5040):

$(6480\times 7)+7!=45360+5040=50400$

The rule consistently produces every term in the sequence. Therefore, the missing value is 168.