# Find the sum of the first 25 terms of an arithmetic sequence whose first term is -9 and whose common difference is 5.?

Relevance
• For an arithmetic sequence:

a₁ = - 9

a₂ = a₁ + d ← where d is the common difference, i.e.: 5

a₃ = a₂ + d = a₁ + 2d

a₃ = a₃ + d = a₁ + 3d

…and you can generalize writing:

a(n) = a₁ + (n - 1).d → for the 25th term, n = 25

a₂₅ = - 9 + (25 - 1).5

a₂₅ = - 9 + 120

a₂₅ = 111

s = a₁ + a₂ + a₃ + a₃ + … + a₂₅

s = (- 9) + (- 4) + (1) + (6) + … + (111)

s = (111) + (106) + (101) + (96) + … + (- 9)

2s = (102) * 25

s = 51 * 25

s = 1275

• -9, -4, 1, 6, ...

an = 5 n - 14

S25 = (25/2)(-9 + 111) = 5100/4 = 1275

• T = -9 = -14 + 5 * 1

T[n] = -14 + 5 * n

T = -14 + 5 * 25 = 125 - 14 = 111

S[n] = (n/2) * (T + T[n])

S = (25/2) * (-9 + 111)

S = (25/2) * (102)

S = 25 * 51

S = 25 * 50 + 25 * 1

S = 1250 + 25

S = 1275

• Apply the formula

Sum to n terms of an AP =

(n/2)*[2a + (n - 1)d]

where n = number of terms = 25

a is first term = -9

d is common difference = 5

Ans: 1275