Example 2 – Sum of A.P.

Step-by-Step Solution:

First term [a = 4]

Common difference [d = 9 − 4 = 5]

Number of terms [n = 18]

Use formula
[Sₙ = \dfrac{n}{2} [2a + (n − 1)d]]

Substitute values:
[S₁₈ = \dfrac{18}{2} [8 + 17 × 5]]

Simplify:
[S₁₈ = 9 [8 + 85] = 9 × 93]

Final answer:
[S₁₈ = 837]

Conclusion: The sum of the first 18 terms is [837].

Step-by-Step Solution:

[a = −3]

[d = 1 − (−3) = 4]

[n = 20]

Apply formula:
[S₂₀ = \dfrac{20}{2} [−6 + 19 × 4]]

Simplify:
[S₂₀ = 10 [−6 + 76] = 10 × 70]

Conclusion: The sum is [700].

Step-by-Step Solution:

[a = 10], [d = 3], [n = 25]

Formula:
[S₂₅ = \dfrac{25}{2} [20 + 24 × 3]]

Simplify:
[S₂₅ = \dfrac{25}{2} [20 + 72]]

[S₂₅ = \dfrac{25}{2} × 92 = 1150]

Conclusion: Required sum is [1150].

Step-by-Step Solution:

[a = 7]

[d = 4 − 7 = −3]

[n = 16]

[S₁₆ = \dfrac{16}{2} [14 + 15(−3)]]

[S₁₆ = 8 [14 − 45] = 8(−31)]

Conclusion: The sum is [−248].

Step-by-Step Solution:

AP: 7, 14, 21, …

[a = 7], [d = 7], [n = 10]

[S₁₀ = \dfrac{10}{2} [14 + 9 × 7]]

[S₁₀ = 5 [14 + 63] = 5 × 77]

Conclusion: The sum is [385].

Step-by-Step Solution:

[a = 2], [d = 5], [n = 30]

[S₃₀ = \dfrac{30}{2} [4 + 29 × 5]]

[S₃₀ = 15 [4 + 145] = 15 × 149]

Conclusion: The sum is [2235].

Step-by-Step Solution:

AP: 1, 3, 5, …

[a = 1], [d = 2], [n = 12]

[S₁₂ = \dfrac{12}{2} [2 + 11 × 2]]

[S₁₂ = 6 [2 + 22] = 6 × 24]

Conclusion: The sum is [144].

Step-by-Step Solution:

[a = 11]

[d = 9 − 11 = −2]

[n = 8]

[S₈ = \dfrac{8}{2} [22 + 7(−2)]]

[S₈ = 4 [22 − 14] = 4 × 8]

Conclusion: The sum is [32].

Step-by-Step Solution:

AP: 2, 4, 6, …

[a = 2], [d = 2], [n = 50]

[S₅₀ = \dfrac{50}{2} [4 + 49 × 2]]

[S₅₀ = 25 [4 + 98] = 25 × 102]

Conclusion: The sum is [2550].

Step-by-Step Solution:

[a = 3], [d = 5]

Let number of terms = [n]

[406 = \dfrac{n}{2} [6 + (n − 1)5]]

[406 = \dfrac{n}{2} (5n + 1)]

[812 = n(5n + 1)]

[5n² + n − 812 = 0]

Solving gives [n = 12]

Conclusion: The required number of terms is [12].