To find the next term in the series [tex]\(28, 24, 13, -5, -30, ?\)[/tex], we need to analyze the pattern of the differences between consecutive terms. Let's go through the process step-by-step:
1. Calculate the differences between consecutive terms:
[tex]\[
\text{First difference: } 24 - 28 = -4
\][/tex]
[tex]\[
\text{Second difference: } 13 - 24 = -11
\][/tex]
[tex]\[
\text{Third difference: } -5 - 13 = -18
\][/tex]
[tex]\[
\text{Fourth difference: } -30 - (-5) = -25
\][/tex]
2. Observe the pattern in the differences:
- The first difference: [tex]\(-4\)[/tex]
- The second difference: [tex]\(-11\)[/tex]
- The third difference: [tex]\(-18\)[/tex]
- The fourth difference: [tex]\(-25\)[/tex]
We notice that these differences follow a linear pattern. Each difference is decreasing by a constant value:
[tex]\[
(-11) - (-4) = -7
\][/tex]
[tex]\[
(-18) - (-11) = -7
\][/tex]
[tex]\[
(-25) - (-18) = -7
\][/tex]
3. Predict the next difference:
Given that each difference decreases by [tex]\(-7\)[/tex], the next difference after [tex]\(-25\)[/tex] will be:
[tex]\[
-25 - 7 = -32
\][/tex]
4. Calculate the next term in the series:
To find the next term, we subtract this next difference from the last term in the series, which is [tex]\(-30\)[/tex]:
[tex]\[
-30 - 32 = -62
\][/tex]
Therefore, the next term in the series [tex]\(28, 24, 13, -5, -30, ?\)[/tex] is [tex]\(-62\)[/tex].
