To find the value of [tex]\( d \)[/tex] such that [tex]\( (2d + 1), (2d - 1) \)[/tex], and [tex]\( (3d + 4) \)[/tex] are in an Arithmetic Sequence (AS), we need to ensure that the middle term is the average of the first and third terms. This means:
[tex]\[ 2(2d - 1) = (2d + 1) + (3d + 4) \][/tex]
Let's break this down step-by-step.
1. Write down the condition for terms in an arithmetic progression (AS):
[tex]\[ 2b = a + c \][/tex]
Here:
[tex]\[ a = 2d + 1 \][/tex]
[tex]\[ b = 2d - 1 \][/tex]
[tex]\[ c = 3d + 4 \][/tex]
2. Substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the AS condition:
[tex]\[ 2(2d - 1) = (2d + 1) + (3d + 4) \][/tex]
3. Simplify both sides of the equation:
[tex]\[ 2(2d - 1) = 4d - 2 \][/tex]
[tex]\[ (2d + 1) + (3d + 4) = 5d + 5 \][/tex]
4. Set the simplified expressions equal to each other:
[tex]\[ 4d - 2 = 5d + 5 \][/tex]
5. Solve for [tex]\( d \)[/tex]:
[tex]\[ 4d - 2 = 5d + 5 \][/tex]
Subtract [tex]\( 4d \)[/tex] from both sides:
[tex]\[ -2 = d + 5 \][/tex]
Subtract 5 from both sides:
[tex]\[ -7 = d \][/tex]
Therefore, the value of [tex]\( d \)[/tex] that ensures [tex]\( (2d + 1), (2d - 1) \)[/tex], and [tex]\( (3d + 4) \)[/tex] are in an Arithmetic Sequence is:
[tex]\[ \boxed{-7} \][/tex]
