Grade 10 Arithmetic Progression

Sequence: A set of numbers arranged in a definite order.

Arithmetic Progression (AP): A sequence in which the difference between any two consecutive terms is constant.

Common Difference (d): The constant difference between consecutive terms. It can be positive, negative, or zero.
d = (any term) - (previous term)

Example: Consider the sequence 2, 5, 8, 11, ...

• 5 - 2 = 3
• 8 - 5 = 3
• 11 - 8 = 3

The difference is consistently 3. So, this is an AP with a common difference d = 3.

Example 2: Consider the sequence 10, 8, 6, 4, ...

• 8 - 10 = -2
• 6 - 8 = -2
• 4 - 6 = -2

This is an AP with a common difference d = -2.

Let the first term of an AP be 'a' (or a1) and the common difference be 'd'.

Terms of the AP:

• 1st term (a1) = a
• 2nd term (a2) = a + d
• 3rd term (a3) = a + 2d
• 4th term (a4) = a + 3d
• ... and so on.

Observe the pattern: The coefficient of 'd' is always one less than the term number.

Formula for the nth term (an or Tn):
an = a + (n - 1)d
where 'a' is the first term, 'd' is the common difference, and 'n' is the term number.

Example: Find the 10th term of the AP 2, 5, 8, 11, ...
Here, a = 2, d = 3, n = 10.

Step 1: Use the formula an = a + (n - 1)d.
a10 = 2 + (10 - 1) * 3

Step 2: Calculate the value.
a10 = 2 + 9 * 3
a10 = 2 + 27
a10 = 29

Result: The 10th term of the AP is 29.

Let Sn be the sum of the first n terms of an AP.

Formula for the Sum of the first n terms (Sn):
Sn = [n / 2] * [2a + (n - 1)d]
where 'a' is the first term, 'd' is the common difference, and 'n' is the number of terms.

Alternative Formula: If the last term (the nth term, an or l) is known, the sum can also be calculated as:
Sn = [n / 2] * [a + l]
where 'a' is the first term, 'l' is the last term, and 'n' is the number of terms.

Example: Find the sum of the first 10 terms of the AP 2, 5, 8, 11, ...
Here, a = 2, d = 3, n = 10.

Step 1: Use the formula Sn = [n / 2] * [2a + (n - 1)d].
S10 = [10 / 2] * [2(2) + (10 - 1) * 3]

Step 2: Perform the calculation.
S10 = 5 * [4 + 9 * 3]
S10 = 5 * [4 + 27]
S10 = 5 * 31
S10 = 155

Result: The sum of the first 10 terms of the AP is 155.