Guys please help me out ASAP
In an arithmetic sequence Sn = n² - 2n.
Use your formula in 2c to determine the value of
(a) T7 =
(b) Tn=

The formula in 2c : Sn - Sn-1 =Tn

Answer:

[tex]T_7 = 11[/tex]

[tex]T_n = 2n- 3[/tex]

Step-by-step explanation:

Given

[tex]S_n = n^2 -2n[/tex]

[tex]T_n = S_n - S_{n-1}[/tex]

Solving (a): T7

Substitute 7 for n in [tex]T_n = S_n - S_{n-1}[/tex]

[tex]T_7 = S_7 - S_{7-1}[/tex]

[tex]T_7 = S_7 - S_6[/tex]

Calculate S7 and S6

[tex]S_n = n^2 -2n[/tex]

[tex]S_7 = 7^2 - 2 * 7 = 35[/tex]

[tex]S_6 = 6^2 - 2 * 6 = 24[/tex]

So:

[tex]T_7 = 35 -24[/tex]

[tex]T_7 = 11[/tex]

Solving (b): Tn

[tex]T_n = S_n - S_{n-1}[/tex]

Where [tex]S_n = n^2 -2n[/tex]

Calculate[tex]S_{n-1}[/tex]

We have:

[tex]S_{n-1} =(n-1)^2 - 2(n-1)[/tex]

Open brackets

[tex]S_{n-1} = n^2 -2n + 1 -2n +2[/tex]

[tex]S_{n-1} = n^2 -2n-2n + 1 +2[/tex]

[tex]S_{n-1} = n^2 -4n + 3[/tex]

So:

[tex]T_n = S_n - S_{n-1}[/tex]

[tex]T_n = n^2 - 2n - (n^2 -4n + 3)[/tex]

[tex]T_n = n^2 - 2n - n^2 +4n - 3[/tex]

Collect Like Terms

[tex]T_n = n^2 - n^2- 2n +4n - 3[/tex]

[tex]T_n = 2n- 3[/tex]

The value of Tn and T7 are 2n - 3 and 11 respectively

Given the formula for calculating the nth term as a function of sum expressed as:

[tex]S_n - S_{n-1} =T_n[/tex]

If Sn = n² - 2n, then;

Sn-1 = (n-1)² - 2(n-1)

Tn =  n² - 2n - [(n-1)² - 2(n-1)]

Tn = n² - 2n- (n² - 2n + 1 - 2n + 2)

Tn = -1 + 2n - 2

Tn = 2n - 3

For T7;

T7 = 2(7) - 3

T7 = 11

Hence the value of Tn and T7 are 2n - 3 and 11 respectively

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