Sure, let's solve for the missing numbers step-by-step.
1. Let's start with the sequence and the operations involved in deriving the missing numbers:
[tex]\[
\begin{array}{cccc}
1 & -3 & -10 & 24 \\
& b & d & \\
1 & c
\end{array}
\][/tex]
2. Finding [tex]\( b \)[/tex]:
Based on the sequence, the relationship given is:
[tex]\[
2 \times 1 + a = b
\][/tex]
From the given array, [tex]\( a = -3 \)[/tex]:
[tex]\[
2 \times 1 + (-3) = b
\][/tex]
Solving that:
[tex]\[
2 - 3 = -1
\][/tex]
Therefore:
[tex]\[
b = -1
\][/tex]
3. Finding [tex]\( c \)[/tex]:
The next relationship is:
[tex]\[
2 \times (-3) + b = c
\][/tex]
Substitute [tex]\( b = -1 \)[/tex]:
[tex]\[
2 \times (-3) + (-1) = c
\][/tex]
Solving that:
[tex]\[
-6 - 1 = -7
\][/tex]
Therefore:
[tex]\[
c = -7
\][/tex]
4. Finding [tex]\( d \)[/tex]:
The final relationship is:
[tex]\[
2 \times (-10) + c = d
\][/tex]
Substitute [tex]\( c = -7 \)[/tex]:
[tex]\[
2 \times (-10) + (-7) = d
\][/tex]
Solving that:
[tex]\[
-20 - 7 = -27
\][/tex]
Therefore:
[tex]\[
d = -27
\][/tex]
So the missing numbers are:
[tex]\[
\begin{aligned}
b &= -1 \\
c &= -7 \\
d &= -27
\end{aligned}
\][/tex]
