Substituting d = 4 back into equation (1), we have a + 4 × 4 = 18. Simplifying gives a + 16 = 18, which leads to a = 18 - 16 = 2.

the fifth time of an ap is 18 and 11th term is 42 find first term

Answer: The first term of the AP is $2$ .

Step-by-step solution

Step 1: Express the fifth term using the general term formula

We know that the $n\text{th}$ term of an arithmetic progression is given by $a_n = a + (n - 1)d$ , where $a$ is the first term and $d$ is the common difference. Substituting $n = 5$ and $a_5 = 18$ , we obtain our first linear equation: $a + 4d = 18$ .

Step 2: Express the eleventh term using the general term formula

Similarly, substituting $n = 11$ and $a_{11} = 42$ into the general formula $a_n = a + (n - 1)d$ , we obtain the second equation: $a + 10d = 42$ .

Step 3: Subtract equation (1) from equation (2) to find d

To eliminate $a$ , we subtract equation $(1)$ from equation $(2)$ . This gives $(10d - 4d) = 42 - 18$ , which simplifies to $6d = 24$ , giving $d = 4$ .

Step 4: Substitute the value of d into equation (1) to find a

Substituting $d = 4$ back into equation $(1)$ , we have $a + 4 \times 4 = 18$ . Simplifying gives $a + 16 = 18$ , which leads to $a = 18 - 16 = 2$ .

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