NCERT Class X Chapter 5: Arithmetic Progression Exercise 5.3 Question 10(i)

NCERT Class X Chapter 5: Arithmetic Progression Exercise 5.3 Question 10(i)

Question:

Show that a1, a2, . . ., an, . . . form an AP where an is defined as below : an = 3 + 4n. Also find the sum of the first 15 terms in each case.

Given:

an = 3 + 4n

To Find:

1. Show that the sequence forms an AP.
2. Find the sum of the first 15 terms.

Formula:

The sum of an arithmetic series is given by: Sn = n2(a1 + an) , where 

n is the number of terms, 

a1 is the first term, and 

an is the last term.

Solution:

To show it's an AP, let's find the difference between consecutive terms:

an+1 - an = [3 + 4(n+1)] - (3 + 4n) 

⇒ an+1 - an = 3 + 4n + 4 - 3 - 4n 

⇒ an+1 - an = 4

Since the difference is constant (4), the sequence is an arithmetic progression.

Now, let's find the sum of the first 15 terms (n=15):

⇒ a1 = 3 + 4(1) = 7

⇒ a15 = 3 + 4(15) = 63

⇒ S15 = 152 (7 + 63) 

⇒ S15 = 152(70) 

⇒ S15 = 15 × 35 

⇒ S15 = 525

Result:

The sequence forms an AP with a common difference of 4. 

The sum of the first 15 terms is 525.

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