How many three-digit numbers are divisible by 7?

Class 10Arithmetic ProgressionsAlgebra — Sequences

Step-by-Step Solution

The smallest three-digit multiple of 7 is 105 and the largest is 994. With

a = 105, d = 7, l = 994 : n = \frac{994 - 105}{7} + 1 = 127 + 1 = 128

. Forgetting the '+ 1' leaves 127.

Answer: B) 128

**Sₙ =

(n /2) \times

[

2 a + (n - 1) d

]**

Or equivalently, **Sₙ =

(n /2) \times (a + l)

**, where

l

is the LAST term (the n-th term).

**Why these are equal**:

l = a + (n - 1) d

, so

a + l = 2 a + (n - 1) d

.

**Example**: Sum of 1, 4, 7, …, 28.
First find

n : a = 1, d = 3, l = 28

. From aₙ = 28:

1 + 3 (n - 1) = 28

→

n = 10

.
Then S₁₀ =

(10/2) (1 + 28) = 5 \times 29 = 145

.

**Special case

(n

natural numbers)**: Sₙ =

n (n + 1) /2

.
**First

n

odd numbers**: Sₙ =

n^{2}

.
**First

n

even numbers**: Sₙ =

n (n + 1)

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