Question

Question #46f4b

Answer 1

286.8 is the 24th term in the sequence.

Explanation:

In an arithmetic sequence we need to know `a and d`.
We know neither of them, and as we are looking for two variables, we will make simultaneous equations.
`T_n = a + (n-1)d`

For `T_5:" " 47.4 = a + 4d " "A`
For`T_10:" " 110.4 = a + 9d " B"`

B-A : `" "63 = 5d`
`" " d = 12.6`

Substitute 12.6 for d in A:

`" " 47.4 = a + 4 xx 12.6`
`" "47.4 - 50.4 = a`

`a = -3`

Now we can write the General Term for this sequence..
`T_n = -3 + (n-1)12.6`

Which term is 286.8??

`-3 + (n-1)12.6 = 286.8`
`-3 + 12.6n -12.6 = 286.8`
`12.6n = 302.4`
`n = 24`

Answer 2

`"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`

`color(blue)("Determine the difference between terms.")`

An arithmetic sequence is of form:

`a"; "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=>s=5`

`=>T_5+5k=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`

`=>s=(5(286.8-47.4))/63`

`s=19" steps from "T_5`

So the term is `T_(5+19)=T_24`