Question

Determine the AP whose third term is 5 and the seventh term is 9.​

Answer 1

$\text{We know that }a_n \text{ term of an AP is given by:}$

$a_n = a + (n-1)d \text{ where }a,d \text{ are the first term and common difference respectively}$

$\implies a_3 = a + (3-1)d = 5$

$\implies a + 2d = 5 \text{ ------ (i)}$

$\implies a_7 = a + (7-1)d = 9$

$\implies a + 6d = 9 \text{ ------ (ii)}$

$\text{subtracting (i) from (ii)}$

$\implies a + 6d - a - 2d = 9 - 5$

$\implies 4d = 4$

$\implies \boxed{d = 1}$

$\text{Substituting in (i)}$

$\implies a + 2(1) = 5$

$\implies a + 2 = 5$

$\implies \boxed{a = 3}$

$\therefore \;\boxed{\text{The required AP is }3,4,5,6....}$