To determine the value of [tex]\( A \)[/tex] given that the sum of any three consecutive digits in this sequence equals 16, let’s analyze the sequence step-by-step.
We are given:
[tex]\[ A \quad _ \quad 9 \quad _ \quad _ \quad _ \quad _ \quad _ \quad _ \quad _ \quad 9 \quad _ \quad 5 \][/tex]
Since the sum of any three consecutive digits is 16, we can set up the following relationships:
1. Considering the first three digits:
[tex]\[ A + \text{blank} + 9 = 16 \][/tex]
Let's denote the blank by [tex]\( x \)[/tex]:
[tex]\[ A + x + 9 = 16 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x = 16 - A - 9 \][/tex]
[tex]\[ x = 7 - A \][/tex]
2. Consider the next three digits involving the digit 9:
[tex]\[ \text{blank} + 9 + \text{blank} = 16 \][/tex]
Denote the blanks around 9 by [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x + 9 + y = 16 \][/tex]
Substituting [tex]\( x \)[/tex] from our previous result, [tex]\( x = 7 - A \)[/tex]:
[tex]\[ (7 - A) + 9 + y = 16 \][/tex]
Solving for [tex]\( y \)[/tex]:
[tex]\[ 16 - 7 + A - 9 + y = 16 \][/tex]
[tex]\[ 7 - 16 + 7 + y = 16 - 16 \][/tex]
[tex]\[ y = A \][/tex]
3. Now, let’s consider another set of three digits that involve [tex]\( A \)[/tex]. Let’s use the consecutive 9's and the provided 5 towards the end of the sequence:
[tex]\[ \text{blank} + 9 + 9 = 16 \][/tex]
[tex]\[ x + 9 + 9 = 16 \][/tex]
Since [tex]\( x \)[/tex] earlier was represented as [tex]\( 7 - A \)[/tex]:
[tex]\[ 7 - A + 9 + 9 = 16 \][/tex]
[tex]\[ 7 + 18 - A = 16 \][/tex]
[tex]\[ 25 - A = 16 \][/tex]
Solving for [tex]\( A \)[/tex]:
[tex]\[ A = 25 - 16 \][/tex]
[tex]\[ A = 9 \][/tex]
Therefore, the value of [tex]\( A \)[/tex] is [tex]\( \boxed{9} \)[/tex].
