Question

The sum of first m terms of an A.P. is $4 m^2 - m$. If its ${n}^{th}$ term is 107. Find the value of n. Also find its ${21}^{st}$ term.

Answer 1

Given the Sum of 1st 'm' terms of the AP as : 4m² - m

⇒ The 1st term of the Given AP can be found by substituting : m = 1

⇒ 1st term of the Given AP = 4(1²) - 1 = 3

⇒ a = 3

The Sum of First Two terms of the Given AP can be found by substituting : m = 2

⇒ The Sum of First two terms = 4(2²) - 2 = 16 - 2 = 14

⇒ The Second term of the Given AP can be found by subtracting the First term from the Sum of First two terms.

⇒ The Second term of the Given AP : 14 - 3 = 11

⇒ The Common Difference of the Given AP = 2nd term - 1st term

⇒  The Common Difference of the Given AP (d) = 11 - 3 = 8

Given that : The nth term is 107

We know that nth term is given by : $T_n = a + (n - 1)d$

⇒ a + (n - 1)d = 107

we got a = 3 and d = 8

substituting we get :

⇒ 3 + (n - 1)8 = 107

⇒ 3 + 8n - 8 = 107

⇒ n = 112/8 = 14

21st Term of the Given AP can be found by nth term formula :

⇒ $T_n = a + (n - 1)d$

⇒ 21st term = a + (21 - 1)d

⇒ 21st term = 3 + 20 × 8 = 163

Answer 2

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Given the Sum of 1st 'm' terms of the AP as : 4m² - m

⇒ The 1st term of the Given AP can be found by substituting : m = 1

⇒ 1st term of the Given AP = 4(1²) - 1 = 3

⇒ a = 3

The Sum of First Two terms of the Given AP can be found by substituting : m = 2

⇒ The Sum of First two terms = 4(2²) - 2 = 16 - 2 = 14

⇒ The Second term of the Given AP can be found by subtracting the First term from the Sum of First two terms.

T_n = a + (n - 1)d

⇒ The Second term of the Given AP : 14 - 3 = 11

⇒ The Common Difference of the Given AP = 2nd term - 1st term

⇒  The Common Difference of the Given AP (d) = 11 - 3 = 8

Given that : The nth term is 107

We know that nth term is given by : Tn= a+(n-1)d

we got a = 3 and d = 8

substituting we get :

⇒ 3 + (n - 1)8 = 107

⇒ 3 + 8n - 8 = 107

⇒ n = 112/8 = 14

⇒ The Second term of the Given AP : 14 - 3 = 11

The Second term of the Given AP : 14 - 3 = 11⇒ The Common Difference of the Given AP = 2nd term - 1st term

The Second term of the Given AP : 14 - 3 = 11⇒ The Common Difference of the Given AP = 2nd term - 1st term21st Term of the Given AP can be found by nth term formula :

⇒ Tn= a+(n-1)d

⇒ 21st term = a + (21 - 1)d

⇒ 21st term = 3 + 20 × 8 = 163

HOPE IT HELPS U :)