NTSE Mathematics – Sum of N Terms
(A concise, beginner‑friendly study guide in Markdown)
1. Simple Explanation
Sum of N terms is the total obtained by adding the first N members of a numerical sequence (or series).
- For an Arithmetic Progression (AP):
[ S_N = a + (a+d) + (a+2d) + \dots + [a+(N-1)d] ] - For a Geometric Progression (GP):
[ S_N = a + ar + ar^2 + \dots + ar^{N-1} ]
The goal is to find a closed‑form expression for (S_N) that depends only on the first term (a), common difference (d) (AP) or common ratio (r) (GP), and the number of terms (N).
2. Key Points
| Concept | Symbol | Formula | Conditions |
|---|---|---|---|
| Arithmetic Series (Sum of AP) | (S_N) | (\displaystyle S_N = \frac{N}{2}\bigl[2a + (N-1)d\bigr]) | (N) positive integer |
| Equivalent: (S_N = \frac{N}{2}(a + a_N)) where (a_N = a+(N-1)d) | |||
| Geometric Series (Sum of GP) | (S_N) | (\displaystyle S_N = a,\frac{1-r^N}{1-r}) | (r \neq 1) |
| Special case (r=1): (S_N = Na) | |||
| Last term of AP | (a_N) | (a_N = a+(N-1)d) | |
| Last term of GP | (a_N) | (a_N = ar^{N-1}) | |
| Derivation technique (GP) | Multiply (S_N) by (r) → (rS_N = ar + ar^2 + \dots + ar^N) → subtract from (S_N) |
3. Derivation of the Formulas
3.1 Arithmetic Series
- Write the series forwards and backwards:
[ S_N = a + (a+d) + \dots + [a+(N-1)d] ] [ S_N = [a+(N-1)d] + [a+(N-2)d] + \dots + a ] - Add term‑by‑term: each pair sums to (2a + (N-1)d).
- There are (N) pairs →
[ 2S_N = N\bigl[2a + (N-1)d\bigr] ] [ \boxed{S_N = \frac{N}{2}\bigl[2a + (N-1)d\bigr]} ]
3.2 Geometric Series
- Let (S_N = a + ar + ar^2 + \dots + ar^{N-1}).
- Multiply by (r):
[ rS_N = ar + ar^2 + \dots + ar^N ] - Subtract:
[ S_N - rS_N = a - ar^N ] [ S_N(1-r) = a(1-r^N) ] - Solve for (S_N):
[ \boxed{S_N = a,\frac{1-r^N}{1-r}}, \quad r \neq 1 ] - If (r = 1), every term equals (a): (S_N = Na).
4. Worked Example
Example 1 – Arithmetic Series
Find the sum of the first 10 terms of the AP: 2, 5, 8, …
- First term (a = 2)
- Common difference (d