Question

Find the missing number in the series: \(2,\,6,\,12,\,20,\,30,\,?\)

• A. \(36\)
• B. \(40\)
• C. \(42\)
• D. \(44\)

Hint:

Check both the differences between consecutive terms and whether each term equals n multiplied by (n+1) for its position n. Testing this product formula for increasing n often reveals the rule instantly.

Answer 1

The Correct Option is C

Concept: Number series problems are often solved by examining the pattern in the differences between consecutive terms.

Step 1: Find the differences between terms. \[ 6-2 = 4 \] \[ 12-6 = 6 \] \[ 20-12 = 8 \] \[ 30-20 = 10 \]

Step 2: Identify the pattern. The differences increase sequentially: \[ 4,\,6,\,8,\,10 \] Next difference: \[ 12 \]

Step 3: Find the next term. \[ 30 + 12 = 42 \] \[ \boxed{42} \]

Answer 2

Concept:

• A number series can sometimes be written directly as a formula in terms of its position n, instead of only tracking the differences between terms.
• Products of two consecutive integers, $n(n+1)$, generate the sequence $2, 6, 12, 20, 30, 42, \ldots$

Step 1: Write each given term using its position number $n$.
$n=1$: $1 \times 2 = 2$
$n=2$: $2 \times 3 = 6$
$n=3$: $3 \times 4 = 12$

Step 2: Check the rule against the remaining given terms.
$n=4$: $4 \times 5 = 20$
$n=5$: $5 \times 6 = 30$
All five terms match $n(n+1)$ exactly, confirming this is the rule for the series.

Step 3: Apply the rule to find the sixth term.
$n=6$: $6 \times 7 = 42$

Final Answer: $42$