Question

The sum of 4 consecutive terms of an AP is 32 and their product is 3465 . Find the numbers
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Answer 1

Answer:

The number are 8 - 2 √2  ,  8 - √2 ,  8 + √2  , 8 + 2 √2  

Step-by-step explanation:

Given as :

The sum of 4 consecutive terms of an AP is 32

Let The 4 consecutive AP are , a - 2 d , a - d , a + d , a + 2 d

According to question

a - 2 d + a - d + a + d + a + 2 d = 32

Or, (a + a + a + a) + ( - 2 d -d + d + 2 d) = 32

Or, 4 a + 0 = 32

∴  a = $\dfrac{32}{4}$

i.e  a = 8

Again

The product of 4 consecutive terms of an AP is 3465

Or, [ ( a - 2 d ) × (a - d) × ( a + d ) × (a + 2 d ) ] = 3465

Or,  [ ( 8 - 2 d ) × (8 - d) × ( 8 + d ) × (8 + 2 d ) ] = 3465

Or, (64 - 4 d² ) ( 64 - d²) = 3465

Or, 4096 - 64 d² - 256 d² + 4 $d^{4}$ = 3465

Or,  4 $d^{4}$ - 320 d² + 631 = 0

Take d² = x

So, 4 x² - 320 x + 631 = 0

Solving this quadratic equation

x = 77.97 , 2.02

Or, d = $\sqrt{78}$ = 8.8

And  d = √2 = 1.414

So, The numbers are

a - 2 d = 8 - 2 √2

a - d = 8 - √2

a + d = 8 + √2

a + 2 d = 8 + 2 √2

Hence, The number are 8 - 2 √2  ,  8 - √2 ,  8 + √2  , 8 + 2 √2   Answer

Answer 2

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