How do you write a rule for the nth term of the arithmetic sequence a_6=43, a_15=107.8?

Jun 26, 2017

${a}_{n} = 7.2 n - 0.2$

Explanation:

$\text{the nth term formula for an arithmetic sequence is}$

• a_n=a_1+(n-1)d

$\text{where " a_1" is the first term and d the common difference}$

${a}_{1} \text{ and d require to be found}$

$\text{using the "color(blue)"nth term formula}$

${a}_{6} = {a}_{1} + 5 d = 43 \to \left(1\right)$

${a}_{15} = {a}_{1} + 14 d = 107.8 \to \left(2\right)$

$\left(2\right) - \left(1\right)$

$\Rightarrow 9 d = 64.8$

$\Rightarrow d = 7.2$

$\text{substitute this value in " (1)" and solve for } {a}_{1}$

${a}_{1} = 43 - \left(5 \times 7.2\right) = 7$

$\Rightarrow {a}_{n} = 7 + \left(n - 1\right) 7.2$

$\Rightarrow {a}_{n} = 7.2 n - 0.2$