# Question 46f4b

Jun 22, 2016

286.8 is the 24th term in the sequence.

#### Explanation:

In an arithmetic sequence we need to know $a \mathmr{and} d$.
We know neither of them, and as we are looking for two variables, we will make simultaneous equations.
${T}_{n} = a + \left(n - 1\right) d$

For ${T}_{5} : \text{ " 47.4 = a + 4d " } A$
For${T}_{10} : \text{ " 110.4 = a + 9d " B}$

B-A : $\text{ } 63 = 5 d$
$\text{ } d = 12.6$

Substitute 12.6 for d in A:

$\text{ } 47.4 = a + 4 \times 12.6$
$\text{ } 47.4 - 50.4 = a$

$a = - 3$

Now we can write the General Term for this sequence..
${T}_{n} = - 3 + \left(n - 1\right) 12.6$

Which term is 286.8??

$- 3 + \left(n - 1\right) 12.6 = 286.8$
$- 3 + 12.6 n - 12.6 = 286.8$
$12.6 n = 302.4$
$n = 24$

Jun 22, 2016

$\text{The term number for 286.8 is } {T}_{24}$

#### Explanation:

Let the number of steps be $s$
Let the 5th term be $a$
Let the difference between terms be $k$

$\textcolor{b l u e}{\text{Determine the difference between terms.}}$

An arithmetic sequence is of form:

$a \text{; "a+k"; "a+2k"; "a+3k"; ........}$

The number of steps between the given terms:

${T}_{5} , {T}_{6} , {T}_{7} , {T}_{8} , {T}_{9} , {T}_{10}$

So there are 5 steps from ${T}_{5}$ to ${T}_{10} \implies s = 5$

$\implies {T}_{5} + 5 k = {T}_{10}$

color(brown)(=>k=(T_10-T_5)/5)color(blue)(" "->" "k=(110.4-47.4)/5=63/5
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Thus for the target term we have:
color(brown)(T_5+sk=286.8" "color(blue)(->" "47.4+63/5s=286.8#

$\implies s = \frac{5 \left(286.8 - 47.4\right)}{63}$

$s = 19 \text{ steps from } {T}_{5}$

So the term is ${T}_{5 + 19} = {T}_{24}$