Question

The sum of the first two terms of an arimethic sequence is 15 and the sum of the next two terms is 43. Write the first 4 terms of the sequence?

Answer 1

the terms are 4, 11, 18, 25

Explanation:

The first equation is: `" "n + n + x = 15`

The second equation is: `" " n + 2x + n + 3x = 43.`

`n` is the first term `x` is the variable of the sequence. so
`n + x" "` is the second term
`n + 2x" "` is the third term
`n + 3x" "` is the fourth term.

Solving for x in terms of

`n + 2x + n + 3x = 43 " "` adding like terms.

`2n + 5x = 43" "` subtracting `5x` from both side sides gives

`2n + 5x -5x = 43 -5x" "` resulting in

`2n = 43 -5x" "` this can be substituted into the first equation.

`n + n + x = 15" "` which is the same as

`2n + x = 15" "` now substitute `2n = 43 -5x` into the question.

`43 -5x + x = 15" "` combine like terms

`43 - 4x = 15` add `4x` to both sides and subtract `15` from both sides

`43 -15 - 4x + 4x = 15-15 + 4x" "` gives.

`28 = 4x " "` Divide both sides by `4` to solve for `x`

`28/4 = (4x)/4` gives

`7 = x " "` This is the variable of the sequence.

Putting `7` in for `x` to solve for `n` the first number in the sequence.

`n + n + 7 = 15" "` combine like terms

`2n + 7 = 15" "` subtract 7 from both sides

`2n + 7 -7 = 15 -7 " "` which gives

`2n = 8 " "` Divide both sides by 2

`(2n)/2 = 8/2` gives

`n = 4` so `4` is the first number of sequence add `7`

`n_2 = 11" " 4 + 7 = 11`

`n_3 = 18" " 11 +7 = 18`

`n_4 = 25" "18 + 7 = 25`

Answer 2

`4" "11" "18" "25`

Explanation:

The terms in an arithmetic sequence can be written in terms of the first term, `a` and the common difference, `d`.

`d` is added to each term to get to the next.

Terms `(n)`: `" "1color(white)(wwwww)2color(white)(wwwwww)3color(white)(wwwwww)4`

Values:`(T_n)" "color(red)(a)color(white)(www)color(red)((a+d))color(white)(ww)color(blue)((a+2d))color(white)(ww)color(blue)((a+3d)))`

We know that the sum of `color(red)(T_1 and T_2 = 15`
We know that the sum of `color(blue)(T_ 3 and T_4 = 43)`

`:.color(red)( a+a+d = 15)color(white)(wwww.w)rarr color(red)(2a +d = 15)...........A`
`:. color(blue)(a+2d + a +3d = 43)" "rarr color(blue)(2a +5d = 43)..........B`

The difference between A and B will give us a value for `4d`

`B - A:color(white)(wwwww) 4d=28`
`color(white)(wwwwwnwwnww) d=7`

We now know that the terms differ by `7` each time.

`:.color(red)( a+a+7 = 15)`
`:.color(red)(2a = 8)`
`:.color(red)(a=4`

Now we can write the first `4` terms:

`4" "11" "18" "25`

Let's check by adding:

`color(red)(4+11 = 15)" and "color(blue)(18+25 =43)`