Arithmetic Sequence

• The nth term (Un)

Un = a + (n - 1) b

• The sum of the first n terms (Sn)

Sn = n/2 × (a + Un)

Sn = n/2 × (2a + (n - 1) b)

a = the first term

b = the common difference

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S₁₀ = 125

U₃ = 5

a = ?

b = ?

S₁₅ = ?

S₁₀ = 125

10/2 × (2a + (10 - 1) b) = 125

5 × (2a + 9b) = 125

2a + 9b = 25 ... eq (1)

U₃ = 5

a + (3 - 1) b = 5

a + 2b = 5 ... eq (2)

To find a and b, use the elimination/substitution

▪︎Find the first term

Eliminate variable b to find a

2a + 9b = 25 (×2)

a + 2b = 5 (×9)

4a + 18b = 50

9a + 18b = 45

___________ -

-5a = 5

a = -1

The first term is -1

▪︎Find the common difference

To find the common difference, substitute a = -1 to eq (2)

a + 2b = 5

-1 + 2b = 5

2b = 5 + 1

2b = 6

b = 3

The common difference is 3

▪︎Find the sum of the first 15 terms

S₁₅ = 15/2 × [2(-1) + (15 - 1)(3)]

S₁₅ = 15/2 × [-2 + (14)(3)]

S₁₅ = 15/2 × (-2 + 42)

S₁₅ = 15/2 × 40

S₁₅ = 15 × 20

S₁₅ = 300

The sum of the first 15 terms is 300.

Hope it helps.