Sure! Let’s solve this step-by-step.
Given:
[tex]\( T_1 = 3 \)[/tex]
[tex]\( T_2 - T_1 = 9 \)[/tex] therefore, [tex]\( T_2 = 3 + 9 = 12 \)[/tex]
[tex]\( T_3 - T_2 = 21 \)[/tex] therefore, [tex]\( T_3 = 12 + 21 = 33 \)[/tex]
Now, we can see that the differences between consecutive terms are increasing. Specifically:
[tex]\( T_2 - T_1 = 9 \)[/tex]
[tex]\( T_3 - T_2 = 21 \)[/tex]
If this pattern continues with a quadratic sequence, the change between differences is constant. We can identify this constant change ([tex]\( D \)[/tex]). Let's proceed to calculate the subsequent terms:
1. Finding [tex]\( D \)[/tex]:
The difference between the differences:
[tex]\( 21 - 9 = 12 \)[/tex]
Thus, each difference increases by [tex]\( D = 12 \)[/tex].
2. Finding [tex]\( T_4 \)[/tex]:
The next difference after [tex]\( 21 \)[/tex] would be:
[tex]\( 21 + 12 = 33 \)[/tex]
Therefore,
[tex]\( T_4 = T_3 + 33 \)[/tex]
[tex]\[ T_4 = 33 + 33 = 66 \][/tex]
3. Finding [tex]\( T_5 \)[/tex]:
The next difference after [tex]\( 33 \)[/tex] would be:
[tex]\( 33 + 12 = 45 \)[/tex]
Therefore,
[tex]\( T_5 = T_4 + 45 \)[/tex]
[tex]\[ T_5 = 66 + 45 = 111 \][/tex]
Thus, it is shown that [tex]\( T_5 = 111 \)[/tex].
