in the 7th term of an ap is 34 and the 13th term is 64 then the 20th term is

Answer: The 20th term of the AP is 99.

Step-by-step solution

Step 1: Set up equations using the nth term formula

We use the standard formula for the $n^{\text{th}}$ term of an arithmetic progression, $a_n = a + (n - 1)d$ , where $a$ is the first term and $d$ is the common difference. Substituting $n = 7$ gives $a + 6d = 34$ , and substituting $n = 13$ gives $a + 12d = 64$ .

Step 2: Solve for the common difference d

Subtracting equation (1) from equation (2) eliminates the variable $a$ . This gives $6d = 30$ , which simplifies to $d = 5$ .

Step 3: Find the first term a

Substitute $d = 5$ into equation (1) to find the first term $a$ . This yields $a + 30 = 34$ , so $a = 4$ .

Step 4: Calculate the 20th term

Using the $n^{\text{th}}$ term formula for $n = 20$ , we substitute $a = 4$ and $d = 5$ . Evaluating $4 + 19 \times 5$ gives $4 + 95 = 99$ .

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