Question

The first two terms of an arithmetic sequence are shown below. 1 2 x+2' x+3' ck to task your mistake. Find the third term of this sequence. Give your answer fully factorised. Watch video Answer > The first two terms of an arithmetic sequence are shown below . 1 2 x + 2 ' x + 3 ' ck to task your mistake . Find the third term of this sequence . Give your answer fully factorised . Watch video Answer >​

Answer 1

in picture

Explanation:

added in the picture

Answer 2

The fully factorized third term of the arithmetic sequence is
`\((3x + 5)/((x + 2)(x + 3))\).`

The arithmetic sequence is defined by the terms
`\((1)/(x + 2)\) and \((2)/(x + 3)\).`

To find the common difference, subtract the first term from the second:

`\[d = (2)/(x + 3) - (1)/(x + 2).\]`

To combine the fractions, find a common denominator:

`\[d = (2(x + 2) - (x + 3))/((x + 2)(x + 3)).\]`

Simplify the numerator:

`\[d = (2x + 4 - x - 3)/((x + 2)(x + 3)) = (x + 1)/((x + 2)(x + 3)).\]`

Now, the third term can be found by adding the common difference to the second term:

`\[T_3 = (2)/(x + 3) + (x + 1)/((x + 2)(x + 3)).\]`

To add the fractions, find a common denominator:

`\[T_3 = (2(x + 2) + (x + 1))/((x + 2)(x + 3)).\]`

Simplify the numerator:

`\[T_3 = (3x + 5)/((x + 2)(x + 3)).\]`

Therefore, the fully factorized third term of the arithmetic sequence is
`\((3x + 5)/((x + 2)(x + 3))\).`

The probable question may be:

The first two terms of an arithmetic sequence are shown below. 1 /x+2, 2/x+3' . Find the third term of this sequence. Give your answer fully factorised.