The 14th term of an A.P. is 96 while the 25th term is 173. Find the product of the 6th and 13th terms

QUESTION: The 14th term of an A.P. is 96 while the 25th term is 173. Find the product of the 6th and 13th terms

Solution

Given:

• The 14th term ($T_{14}$) is 96.
• The 25th term ($T_{25}$) is 173.
• The formula for the nth term of an A.P. is: $$T_n = a + (n - 1) d$$

Step 1: Setting up equations

For the 14th term:

$$a + 13d = 96$$

For the 25th term:

$$a + 24d = 173$$

Step 2: Solving for $a$ and $d$

Subtract the first equation from the second:

$$(a + 24d) - (a + 13d) = 173 - 96$$$$a + 24d - a - 13d = 77$$$$11d = 77$$$$d = 7$$

Substituting $d = 7$ into the first equation:

$$a + 13(7) = 96$$$$a + 91 = 96$$$$a =$$

Step 3: Finding the 6th and 13th terms

• $T_6 = a + 5d = 5 + 5(7) = 5 + 35 = 40$
• $T_{13} = a + 12d = 5 + 12(7) = 5 + 84 = 89$

Step 4: Calculating the product

$$T_6 \times T_{13} = 40 \times 89 = 3560$$