Finding the Missing Number in the Series

This question asks us to identify the missing number in the given series: 6, 35, 144, 435, 872, ?

To find the missing number, we need to analyze the sequence and discover the underlying pattern or rule that connects the numbers.

Let's examine the relationship between consecutive terms:

First term = 6

Second term = 35

Third term = 144

Fourth term = 435

Fifth term = 872

Sixth term = ?

Discovering the Series Pattern

Let's try to find a pattern involving multiplication and addition or subtraction between consecutive terms.

• From 6 to 35: $6 \times 5 + 5 = 30 + 5 = 35$
• From 35 to 144: $35 \times 4 + 4 = 140 + 4 = 144$
• From 144 to 435: $144 \times 3 + 3 = 432 + 3 = 435$
• From 435 to 872: $435 \times 2 + 2 = 870 + 2 = 872$

It appears there is a consistent pattern:

Each term after the first is obtained by multiplying the previous term by a decreasing integer (starting from 5) and then adding the same decreasing integer to the result.

Let $T_n$ be the n-th term in the series. The pattern can be described as:

• $T_2 = T_1 \times 5 + 5$
• $T_3 = T_2 \times 4 + 4$
• $T_4 = T_3 \times 3 + 3$
• $T_5 = T_4 \times 2 + 2$

In general, for $n > 1$, the pattern is $T_n = T_{n-1} \times (7-n) + (7-n)$. Let's verify this using the terms given:

• For $n=2$: $T_2 = T_1 \times (7-2) + (7-2) = 6 \times 5 + 5 = 30 + 5 = 35$ (Correct)
• For $n=3$: $T_3 = T_2 \times (7-3) + (7-3) = 35 \times 4 + 4 = 140 + 4 = 144$ (Correct)
• For $n=4$: $T_4 = T_3 \times (7-4) + (7-4) = 144 \times 3 + 3 = 432 + 3 = 435$ (Correct)
• For $n=5$: $T_5 = T_4 \times (7-5) + (7-5) = 435 \times 2 + 2 = 870 + 2 = 872$ (Correct)

Calculating the Missing Number

To find the next number in the series (the sixth term, $T_6$), we apply the same pattern for $n=6$:

• For $n=6$: $T_6 = T_5 \times (7-6) + (7-6)$
• $T_6 = 872 \times 1 + 1$
• $T_6 = 872 + 1$
• $T_6 = 873$

Therefore, the missing number in the series is 873.

Summary of the Pattern and Calculation

The pattern followed is to multiply the previous term by a decreasing factor (5, 4, 3, 2, 1) and add the same factor. The next term is calculated as:

$872 \times 1 + 1 = 873$

The completed series is 6, 35, 144, 435, 872, 873.