Answer:
[tex]T_n = 2n - 1 + T_{n-1}[/tex]
Step-by-step explanation:
Given
[tex]Sequence: -5,-2,3,10,19[/tex]
Required
Find the nth term
We have:
[tex]T_1 = -5[/tex]
[tex]T_2 = -2 = 3 - 5 = 3 + T_1[/tex]
[tex]T_3 = 3 = 5 - 2 = 5 + T_2[/tex]
[tex]T_4 = 10 = 7 + 3 = 7 +T_3[/tex]
[tex]T_5 = 19 = 9 + 10 = 9 + T_4[/tex]
This gives:
[tex]T_2 = 3 + T_1 = 4 - 1 + T_1[/tex]
[tex]T_3 = 5 + T_2 = 6 - 1 + T_2[/tex]
[tex]T_4 = 7 +T_3 = 8 - 1 + T_3[/tex]
[tex]T_5 = 9 + T_4 = 10 - 1 + T_4[/tex]
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Solving further, we have:
[tex]T_2 = 4 - 1 + T_1 = 2 * 2 - 1 + T_1[/tex]
[tex]T_3 = 6 - 1 + T_2 = 2 * 3 + T_2[/tex]
[tex]T_4 = 8 - 1 + T_3 = 2 * 4 - 1 + T_3[/tex]
[tex]T_5 = 10 - 1 + T_4 = 2 * 5 - 1 + T_4[/tex]
Following the above sequence: the nth term is:
[tex]T_n = 2n - 1 + T_{n-1}[/tex]
