Respuesta :
Answer:
The first three terms of the original sequence are 9, 14, and 22.
Step-by-step explanation:
From the question,
The first difference of the sequence is 5, 8, 11, 14,... That is,
[tex]T_{2} - T_{1} = 5\\T_{3} - T_{2} = 8\\T_{4} - T_{3} = 11\\T_{5} - T_{4} = 14[/tex]
Where [tex]T_{1}[/tex] is the first term of the original sequence
[tex]T_{2}[/tex] is the second term of the original sequence
[tex]T_{3}[/tex] is the third term of the original sequence etc.
Also, from the question, the sum of the first two terms of the original sequence is 23; that is,
[tex]T_{1} + T_{2} = 23[/tex]
Now, we can find [tex]T_{1}[/tex] and [tex]T_{2}[/tex] by solving the following equations simultaneously
[tex]T_{2} - T_{1} = 5[/tex] .......... (1)
[tex]T_{1} + T_{2} = 23[/tex] ......... (2)
From equation (1)
[tex]T_{2} - T_{1} = 5[/tex]
Then,
[tex]T_{2} = 5 + T_{1}[/tex] .......... (3)
Substitute the value of [tex]T_{2}[/tex] into equation (2)
Then, [tex]T_{1} + T_{2} = 23[/tex] becomes
[tex]T_{1} + 5 + T_{1} = 23\\[/tex]
Then,
[tex]2T_{1} + 5 = 23\\2T_{1} = 23 - 5\\2T_{1} = 18[/tex]
[tex]T_{1} = \frac{18}{2} \\T_{1} = 9[/tex]
Hence, the first term of the original sequence is 9
Now, substitute the value of [tex]T_{1}[/tex] into equation (3)
Then, [tex]T_{2} = 5 + T_{1}[/tex] become
[tex]T_{2} = 5 + 9\\[/tex]
∴[tex]T_{2} = 14[/tex]
Hence, the second term of the original sequence is 14
The third term of the original sequence is given by
[tex]T_{3} - T_{2} = 8[/tex]
Then,
[tex]T_{3} = 8 + T_{2}[/tex]
[tex]T_{3} = 8 + 14\\T_{3} = 22[/tex]
Hence, the third term of the original sequence is 22.
Hence, the first three terms of the original sequence are 9, 14, and 22.
