Find the first term and common difference for each of the A.P.

Описание: Find the first term and common difference for each of the A.P.
(1) 5, 1, -3, -7, . . .
(2) 0.6, 0.9, 1.2, 1.5, . . .
(3) 127, 135, 143, 151, . . .
(4) 1 /4 ,3/ 4 , 5/4 ,7/ 4 , . . .
Finding the First Term & Common Difference in an Arithmetic Progression | Step-by-Step Solution

In this video, we will learn how to find the first term \( a \) and common difference \( d \) for a given arithmetic progression (A.P.). Watch till the end for a step-by-step explanation with solved examples!

📌 *Question:*
Find the first term and common difference for each of the following A.P.:

(1) \( 5, 1, -3, -7, \dots \)
(2) \( 0.6, 0.9, 1.2, 1.5, \dots \)
(3) \( 127, 135, 143, 151, \dots \)
(4) \( \frac{1}{4}, \frac{3}{4}, \frac{5}{4}, \frac{7}{4}, \dots \)

🔢 *Topics Covered:*
✔ Understanding arithmetic progressions (A.P.)
✔ How to identify the first term \( a \)
✔ Finding the common difference \( d \)
✔ Step-by-step explanation with solved examples

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*Solution:*
The general formula for an arithmetic progression (A.P.) is:
\[
a, a + d, a + 2d, a + 3d, \dots
\]
The *first term* is \( a \) (the first number in the sequence).
The *common difference* is found by:
\[
d = a_2 - a_1
\]

1️⃣ *For* \( 5, 1, -3, -7, \dots \)
First term: \( a = 5 \)
Common difference:
\[
d = 1 - 5 = -4
\]
✅ *\( a = 5, d = -4 \)*

2️⃣ *For* \( 0.6, 0.9, 1.2, 1.5, \dots \)
First term: \( a = 0.6 \)
Common difference:
\[
d = 0.9 - 0.6 = 0.3
\]
✅ *\( a = 0.6, d = 0.3 \)*

3️⃣ *For* \( 127, 135, 143, 151, \dots \)
First term: \( a = 127 \)
Common difference:
\[
d = 135 - 127 = 8
\]
✅ *\( a = 127, d = 8 \)*

4️⃣ *For* \( \frac{1}{4}, \frac{3}{4}, \frac{5}{4}, \frac{7}{4}, \dots \)
First term: \( a = \frac{1}{4} \)
Common difference:
\[
d = \frac{3}{4} - \frac{1}{4} = \frac{2}{4} = \frac{1}{2}
\]
✅ *\( a = \frac{1}{4}, d = \frac{1}{2} \)*

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*Final Answer:*
✅ *First term & common difference:*
(1) *\( a = 5, d = -4 \)*
(2) *\( a = 0.6, d = 0.3 \)*
(3) *\( a = 127, d = 8 \)*
(4) *\( a = \frac{1}{4}, d = \frac{1}{2} \)*

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