Check whether 301 is a term of the list of numbers 5, 11, 17, 23, . . .
Step-by-Step Solution
Key Concept: For an arithmetic progression (AP) with first term \(a\) and common difference \(d\), a number \(x\) is a term of the AP iff \((x-a)\) is divisible by \(d\). This follows from the nth‑term formula \(a_n = a + (n-1)d\).
1. Identify the first term and common difference of the given AP.
\[ a = 5 \]
The successive terms increase by \(6\) (\(11-5 = 6\)), so \( d = 6 \).
2. Write the general (nth) term of the AP.
\[ a_n = a + (n-1)d = 5 + (n-1)\times 6 \]
3. Set the given number \(301\) equal to the nth term and solve for \(n\).
\[ 301 = 5 + (n-1)\times 6 \]
\[ 301 - 5 = (n-1)\times 6 \]
\[ 296 = 6(n-1) \]
\[ n-1 = \frac{296}{6} = 49\frac{1}{3} \]
4. Since \(n-1\) is not an integer, \(n\) is not a natural number. Hence \(301\) cannot be expressed as \(a_n\) for any integer \(n\ge 1\).
5. Alternatively, check the divisibility condition:
\[ 301 - 5 = 296 \]
\[ 296 \div 6 = 49\frac{1}{3} \] (not an integer). Therefore \(301\) is not a term of the AP.
6. Conclude: 301 is not a term of the given list.
Correct Answer: No, 301 is not a term of the AP 5, 11, 17, 23, ...