Question

Find the next number in the series: 1, 4, 10, 20, 35, 56, 84, ?

• A. 240
• B. 120
• C. 140
• D. 180

Hint:

The sequence is formed using triangular numbers: \(1,4,10,20,35,56,84 = \binom{3}{3},\binom{4}{3},\binom{5}{3},\ldots\)

Answer 1

The Correct Option is B

Step 1: Find the differences between consecutive terms.
\[ 4-1=3,\quad 10-4=6,\quad 20-10=10 \] \[ 35-20=15,\quad 56-35=21,\quad 84-56=28 \] Thus, the differences are: \[ 3,\;6,\;10,\;15,\;21,\;28 \]

Step 2: Observe the pattern in the differences.
The successive differences are: \[ 6-3=3,\quad 10-6=4,\quad 15-10=5 \] \[ 21-15=6,\quad 28-21=7 \] Hence the next increase should be \(8\). Therefore, the next difference is: \[ 28+8=36 \]

Step 3: Find the next term.
\[ 84+36=120 \] Hence, the next number is {120}