Question

The sum of first n. terms of an A.P. is 5n² + 3n.
a) Find the A.P.
b) Find its general term
c) If its mth term is 168, Find the value of m.​

Answer 1

Given :

• The sum of first n terms of an A.P. is 5n² + 3n.

To find :

a) Find the A.P.

b) Find its general term

c) If its mth term is 168, Find the value of m.

Solution :

★ 5n² + 3n

• Take n = 1

→ 5 × (1)² + 3 × 1

→ 8 = S₁

• Take n = 2

→ 5 × (2)² + 3 × 2

→ 5 × 4 + 6

→ 20 + 6 = 26 = S₂

• Take n = 3

→ 5 × (3)² + 3 × 3

→ 5 × 9 + 9

→ 45 + 9 = 54 = S₃

• Take n = 4

→ 5 × (4)² + 3 × 4

→ 5 × 16 + 12

→ 80 + 12 =92 = S₄

• Take n = 5

→ 5 × (5)² + 3 × 5

→ 5 × 25 + 15

→ 125 + 15 = 140 = S₅

8, 26, 54, 92, 140... are the A.P

General term

→ an = a + (n - 1)d

where,

• a = first term
• n = number of term
• d = common difference

→ Sum of first term = a₁ = first term

•°• S₁ = a₁ = 8

→ S₂ - S₁ = a₂

→ 26 - 8 = 18

•°• a₂ = 18

→ S₃ - S₂ = a₃

→ 54 - 26 = 20

•°• a₃ = 28

→ S₄ - S₃ = a₄

→ 92 - 54 = 38

•°• a₄ = 38

→ S₅ - S₄ = a₅

→ 140 - 92 = 48

•°• a₅ = 48

8, 18, 28, 38, 48....are general terms of A.P

• Given = mth = 168
• Common difference (d) = a₂ - a₁
• d = 18 - 8 = 10

→ an = a + (n - 1)d

→ 168 = 8 + (m - 1) × 10

→ 168 = 8 + 10m - 10

→ 168 = 10m - 2

→ 168 + 2 = 10m

→ 170 = 10m

→ m = 170/10

→ m = 17

•°• m = 17

Answer 2

Given:-

• Sum of first n terms of an A.P. = 5n²+3n.

To Find:-

• The A.P.

• It's General term

• if it's mth term is 168, then m = ?

Formula used:-

• $a_n=a + (n - 1)d$

Solution:-

a) Find the A.P.

Aa given 5n² + 3n,

Putting n = 1

$\implies\:\:\:\:\:S_1=5n^2 + 3n$

$\implies\:\:\:\:\:S_1=5(1)^2 + 3(1)$

$\implies\:\:\:\:\:S_1=5+3$

$\implies\:\:\:\:\:S_1=8$

Putting n = 2

$\implies\:\:\:\:\:S_2=5n^2 + 3n$

$\implies\:\:\:\:\:S_2=5(2)^2 + 3(2)$

$\implies\:\:\:\:\:S_2=20+6$

$\implies\:\:\:\:\:S_2=26$

Putting n = 3

$\implies\:\:\:\:\:S_3=5n^2 + 3n$

$\implies\:\:\:\:\:S_3=5(3)^2 + 3(3)$

$\implies\:\:\:\:\:S_3=45+9$

$\implies\:\:\:\:\:S_3=54$

Putting n = 4

$\implies\:\:\:\:\:S_4=5n^2 + 3n$

$\implies\:\:\:\:\:S_4=5(4)^2 + 3(4)$

$\implies\:\:\:\:\:S_4=80+12$

$\implies\:\:\:\:\:S_4=92$

Now, to find the A.P.

• $a_1 = S_1= 8$

• $a_2 = S_2-S_1= 26-8 = 18$

• $a_3 = S_3-S_2= 54-26=28$

• $a_4 = S_4-S_3= 92-54=38$

Hence, the A.P. we got is 8, 18, 28, 38,. . . .

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b) General term

General term = $a_n$

So, the General terms are,

$a_1, a_2, a_3, a_4$

$8, 18, 28, 38$

Hence, 8, 18, 28, 38, are the general terms.

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c) If its mth term is 168, Find the value of m.

Here we get an A.P i.e. 8, 18, 28, 38,

where,

• a = 8

• d = 18 - 8 = 10

• n = m

• $a_m=168$

Using Formula,

$\implies\:a_n=a + (n - 1)d$

$\implies\:a_m=8 + (m - 1)10$

$\implies\:168=8 + 10m-10$

$\implies\:168=10m-2$

$\implies\:168+2=10m$

$\implies\:m=\dfrac{170}{10}$

${\boxed{\implies\:m=17}}$

Hence, The Value of m is 17.

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