HOW TO FIND THE FIRST TERM AND COMMON DIFFERENCE OF AN ARITHMETIC SEQUENCE

Problem 1 :

Given t₁ = 7 and t₁₃ = 35. Find d and S₁₃.

Solution :

t₁₃ = 35

t₁ + (13 - 1)d = 35

t₁ + 12d = 35

Substitute t₁ = 7.

7 + 12d = 35

Subtract 7 from both sides.

12d = 28

Divide both sides by 12.

d = 7/3

Formula to find the sum of first n terms of an arithmetic sequence :

S_n = (n/2)[2t₁ + (n - 1)d]

Substitute n = 13, t₁ = 7 and d = 7/3.

Problem 2 :

Given t₁₂ = 37 and d = 3. Find t₁ and S₁₂.

Solution :

t₁₂ = 37

t₁ + (12 - 1)d = 37

t₁ + 11d = 37

Substitute d = 3.

t₁ + 11(3) = 37

t₁ + 33 = 37

Subtract 33 from both sides.

t₁ = 4

Formula to find the sum of first n terms of an arithmetic sequence :

S_n = (n/2)[2t₁ + (n - 1)d]

Substitute n = 12, t₁ = 4 and d = 3.

= (12/2)[2 ⋅ 4 + (12 - 1)(3)]

= 6[8 + (11)(3)]

= 6(8 + 33)

= 6(41)

= 246

Problem 3 :

Given t₃ = 15 and S₁₀ = 125. Find t₁, d and t₁₀.

Solution :

t₃ = 15

t₁ + (3 - 1)d = 15

t₁ + 2d = 15 ----(1)

S₁₀ = 125

(10/2)[2t₁ + (10 - 1)d] = 125

5[2t₁ + 9d] = 125

Divide both sides by 5.

2t₁ + 9d = 25 ----(2)

(2) - 2(1) :

5d = -5

Divide both sides by 5.

d = -1

Substitute d = -1 in (1).

t₁ + 2(-1) = 15

t₁ + -2 = 15

Add 2 to both sides.

t₁ = 17

Formula to find n^th term of an arithmetic sequence :

t_n  =  t₁ + (n - 1)d

Substitute n = 10, t₁ = 17 and d = -1.

t₁₀ = 17 + (10 - 1)(-1)

t₁₀ = 17 + 9(-1)

t₁₀ = 17 - 9

t₁₀ = 8

Problem 4 :

Given d = 5 and S₉ = 75. Find t₁, d and t₉.

Solution :

S₉ = 75

(9/2)[2t₁ + (9 - 1)(5)] = 75

(9/2)[2t₁ + 8(5)] = 75

(9/2)[2t₁ + 40] = 75

Multiply both sides by 2/9.

2t₁ + 40 = 150/9

2t₁ + 40 = 50/3

Subtract 40 from both sides.

2t₁ = 50/3 - 40

2t₁ = (50 - 120)/3

2t₁ = -70/3

Divide both sides by 2.

t₁ = -35/3

Formula to find n^th term of an arithmetic sequence :

t_n  =  t₁ + (n - 1)d

Substitute n = 9, t₁ = -35/3 and d = 5.

t₉ = -35/3 + (9 - 1)(5)

t₉ = -35/3 + 8(5)

t₉ = -35/3 + 40

t₉ = (-35 + 120)/3

t₉ = 85/3

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