Question

the sum of the first three terms of an ap is 42 and the product of the first and third is 52. find the first term and the common difference.​

Answer 1

Answer:

The first term of the AP is 2 and the common difference = 12

Step-by-step explanation:

Let "d" be the common difference of the AP

We can denote the three terms as:

x - d, x, x + d

Given: Sum of three terms = 42

=> (x - d) + x + (x - d) = 42

=> 3x = 42

=> x = 14

Given: The product of the first and third term = 52

=> (x - d)(x + d) = 52

=> x² - d² = 52

But x = 14

=> 14² - d² = 52

=> 196 - d² = 52

=> d² = 196 - 52

=> d² = 144

=> d = 12 [We are assuming that the common difference is positive]

The three terms of the AP are:

x - d, x, x + d

= 14-12, 14, 14+12

= 2, 14, 26

Answer: The first term of the AP is 2 and the common difference = 12

Verification:

Sum of first three terms = 2 + 14 + 26 = 42 √

Product of first and third term = 2*26 = 52 √

Answer 2

Let the first term of the AP be (a + d), second term be (a) and third term be (a - d).

According To Question ;

(a + d) + a + (a - d) = 42

3a = 42

a = 14

Also, (a + d) (a - d) = 52

a² - d² = 52

Putting value of a in above equation;

(14)² - d² = 52

196 - d² = 52

d² = 144

d = 12

FIRST TERM = (a - d) = (14 - 12) = 2

COMMON DIFFERENCE = d = 12

Hope it helps..