Applying the formula a_m = a' + (m - 1)d' gives 78 = 5 + 6(m - 1), which leads to 6(m - 1) = 73, so m - 1 = (73)/(6) and m = (79)/(6) = 13.17. Since m must be a positive integer, 78 is not a term of this AP.

Applying the formula a_m = a' + (m - 1)d' gives 78 = 5 + 6(m - 1), which leads to 6(m - 1) = 73, so m - 1 = (73)/(6) and m = (79)/(6) = 13.17. Since m must be a positive integer, 78 is not a term of this AP.

which term of the ap 381318 is 78 is 78 also a term of the AP 5 11 17 23

Answer: In the AP $3, 8, 13, 18, \dots$ , $78$ is the $16^{\text{th}}$ term. In the AP $5, 11, 17, 23, \dots$ , $78$ is not a term.

Step-by-step solution

Step 1: Identify parameters for the first AP

For the first AP: $3, 8, 13, 18, \dots$ , the first term is $a = 3$ . The common difference is found by subtracting consecutive terms: $d = 8 - 3 = 5$ . We let the $n$ -th term be $a_n = 78$ .

Step 2: Find the term number for the first AP

We use the $n$ -th term formula $a_n = a + (n - 1)d$ and substitute the values $a = 3$ , $d = 5$ , and $a_n = 78$ . Subtracting $3$ from both sides gives $5(n - 1) = 75$ , which simplifies to $n - 1 = 15$ , giving $n = 16$ .

Step 3: Set up equation for the second AP

For the second AP: $5, 11, 17, 23, \dots$ , the first term is $a' = 5$ and the common difference is $d' = 11 - 5 = 6$ . Suppose $78$ is the $m$ -th term of this progression.

Step 4: Solve for m and check validity

Applying the formula $a_m = a' + (m - 1)d'$ gives $78 = 5 + 6(m - 1)$ , which leads to $6(m - 1) = 73$ , so $m - 1 = \frac{73}{6}$ and $m = \frac{79}{6} = 13.17$ . Since $m$ must be a positive integer, $78$ is not a term of this AP.

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