Question

ಒಂದು ಸಮಾಂತರ ಶ್ರೇಡಿಯ ಮೋದಲ 3 ಪದಗಳ ಮೊತ್ತ 33 ಆಗಿದೆ ಮೋದಲ ಮತ್ತು ಮೂರನೆಯ ಪದಗಳ ಗುಣ ಅದರ 2 ನೇ ಪದಕ್ಕಿಂತ 29 ಹೆಚ್ಚಾಗಿದರೆ ಸಮಾಂತರ ಶ್ರೇಡಿ

Answer 1

Given:

ಒಂದು ಸಮಾಂತರ ಶ್ರೇಡಿಯ ಮೋದಲ 3 ಪದಗಳ ಮೊತ್ತ 33 ಆಗಿದೆ ಮೋದಲ ಮತ್ತು ಮೂರನೆಯ ಪದಗಳ ಗುಣ ಅದರ 2 ನೇ ಪದಕ್ಕಿಂತ 29 ಹೆಚ್ಚಾಗಿದರೆ ಸಮಾಂತರ ಶ್ರೇಡಿ

The sum of the first three terms of an A.P is 33. If the product of the first and the third term exceeds the second term by29, find the A.P

To find:

The A.P.

Solution:

We know the sum of n term of an A.P. is given as:

$\boxed{\bold{S_n = \frac{n}{2}[2a + (n-1)d] }}$  

where Sₙ = last term, a = first term, n = no. of terms and d = common difference

Let us consider the terms of the A.P. as follows:

a, a + d, a + 2d, a + 3d . . .  

As per the question, we get

$S_3 = 33$

$\implies \frac{3}{2}[2a + (3-1)d] = 33$

$\implies \frac{3}{2}[2a + 2d] = 33$

$\implies \frac{3}{2}\times 2[a + d] = 33$

$\implies a + d = 11$ . . .  (1)  

 

Also, we know the nth term of an A.P. is given as:

$\boxed{\bold{a_n = a+ (n-1)d}}$

where aₙ = last term, a = first term, n = no. of terms and d = common difference

Therefore,

$a_3 = a + (3-1)d = a + 2d$

$a_2 = a + (2-1)d = a + d$

As given that in the AP the product of the first and the third term exceeds the second term by 29, so the equation will be,

$(a_1 \times a_3) - a_2 = 29$

$\implies [a \times (a +2d)] - [a + d] = 29$

$\implies a^2 + 2ad - [a + d] = 29$

on substituting from (1), we get

$\implies a^2 + 2a(11 - a) - 11 = 29$

$\implies a^2 + 22a - 2a^2 = 40$

$\implies 22a - a^2 = 40$  

$\implies a^2 - 22a +40 = 0$

$\implies a^2 - 20a - 2a +40 = 0$

$\implies a(a -20) - 2(a - 20) = 0$

$\implies (a -20) (a- 2)= 0$

$\implies a = 2 \:or\:20$

Therefore,

When a = 2,

⇒ d = 11 - a = 11 - 2 = 9

When a = 20,

⇒ d = 11 - 20 = -9

Thus, the possible A.P.s will be as follows:

When a = 2 and d = 9:

a = 2

a + d = 2 + 9 = 11

a + 2d = 2 + 2(9) = 2 + 18 = 20

a + 3d = 2 + 3(9) = 2 + 27 = 29

→ 2, 11, 20, 29, . . . ←

When a = 20 and d = -9:

a = 20

a + d = 2 - 9 = -7

a + 2d = 2 + 2(-9) = 2 - 18 = -16

a + 3d = 2 + 3(-9) = 2 - 27 = -25

→ 20, -7, -16, -25, . . . ←

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