Substitute d = 5 into equation (1). We get a + 6 × 5 = 34, which gives a + 30 = 34, so a = 4.

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

Answer: 99

Step-by-step solution

Step 1: Set up equations using the general term formula

Let the first term of the AP be $a$ and the common difference be $d$ . The $n$ -th term of an AP is given by the formula $a_n = a + (n - 1)d$ . Using the given values for the 7th and 13th terms, we can write two linear equations.

Step 2: Form linear equations from the given terms

For the 7th term, $n = 7$ , giving $a + (7 - 1)d = a + 6d = 34$ . For the 13th term, $n = 13$ , giving $a + (13 - 1)d = a + 12d = 64$ . We now have a pair of linear equations in $a$ and $d$ .

Step 3: Solve for the common difference d

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

Step 4: Substitute d to find the first term a

Substitute $d = 5$ into equation (1). We get $a + 6 \times 5 = 34$ , which gives $a + 30 = 34$ , so $a = 4$ .

Step 5: Calculate the 20th term

To find the 20th term, substitute $n = 20$ , $a = 4$ , and $d = 5$ into the formula $a_n = a + (n - 1)d$ . This gives $a_{20} = 4 + 19 \times 5 = 4 + 95 = 99$ .

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