Question

Complete the series: \(285,\,253,\,221,\,189,\, ?\)

• A. \(165\)
• B. \(157\)
• C. \(169\)
• D. \(145\)

Hint:

Check whether the difference between consecutive terms stays exactly the same throughout the series, using more than one pair of terms to be sure. If it does, this is an arithmetic progression, and the nth-term formula can give you any term directly instead of subtracting step by step.

Answer 1

The Correct Option is B

Concept: A number series often follows a specific pattern such as a constant difference, constant ratio, or another arithmetic rule. Here we check the difference between consecutive terms.

Step 1: Find the difference between consecutive terms. \[ 285 - 253 = 32 \] \[ 253 - 221 = 32 \] \[ 221 - 189 = 32 \] Thus, the difference between each term is constant.

Step 2: Identify the pattern. Since each term decreases by \(32\), the series follows an arithmetic progression with common difference \[ d = -32 \]

Step 3: Find the next term. \[ 189 - 32 = 157 \] \[ \boxed{157} \]

Answer 2

Concept:

• A sequence with a constant difference between consecutive terms is an arithmetic progression (AP), and any term of an AP can be found directly using the general formula $T_n = a + (n-1)d$, without listing out every term one by one.
• This is especially useful for confirming that a pattern is genuine and for finding a term without computing all the terms before it.

Step 1: Identify the first term and the common difference, checking it against more than one pair of terms.
$a = T_1 = 285$
$d = T_2 - T_1 = 253 - 285 = -32$
Check: $T_3 - T_2 = 221 - 253 = -32$, the same value, confirming this is genuinely an AP.

Step 2: Write the general term formula for the series.
$T_n = a + (n-1)d = 285 + (n-1)(-32)$

Step 3: Identify which term is being asked for.
$189$ is the fourth term, $T_4$, so the missing value is the fifth term, $T_5$.

Step 4: Substitute $n = 5$ into the formula.
$T_5 = 285 + (5-1)(-32) = 285 + 4(-32) = 285 - 128$
$T_5 = 157$

Final Answer: $157$