Question

the sum of 1st three terms of an arithmetic progression is 33. if the product of 1st and 3rd term exceeds the 2nd tenm by 29 find ap​

Answer 1

Solution :

$\bf{\red{\underline{\bf{Given\::}}}}$

The sum of 1st three terms of an arithmetic progression is 33. If the product of 1st term and 3rd term exceeds the 2nd term by 29.

$\bf{\red{\underline{\bf{To\:find\::}}}}$

The A.P.

$\bf{\red{\underline{\bf{Explanation\::}}}}$

Let the three terms of an A.P. are;

• (a-d)
• a
• (a+d)

A/q

$\longrightarrow\sf{(a-d)+a+(a+d)=33}\\\\\longrightarrow\sf{a\cancel{-d}+a+a\cancel{+d}=33}\\\\\longrightarrow\sf{3a=33}\\\\\longrightarrow\sf{a=\cancel{\dfrac{33}{3} }}\\\\\longrightarrow\sf{\green{a=11}}$

&

$\longrightarrow\sf{(a-d)(a+d)=a+29}\\\\\longrightarrow\sf{a^{2} \cancel{+ad-ad} -d^{2} =a+29}\\\\\longrightarrow\sf{a^{2} -d^{2} =a+29}\\\\\longrightarrow\sf{(11)^{2} -d^{2} =11+29\:\:\:\:[\therefore\:a=11]}\\\\\longrightarrow\sf{121-d^{2} =40}\\\\\longrightarrow\sf{-d^{2} =40-121}\\\\\longrightarrow\sf{\cancel{-}d^{2} =\cancel{-}81}\\\\\longrightarrow\sf{d=\pm\sqrt{81} }\\\\\longrightarrow\sf{\green{d=\pm9}}$

Now;

$\underline{\underline{\bf{The\:Arithmetic\:Progression\::}}}}}$

For +9

$\bullet\sf{(a-d)=11-9=2}\\\bullet\sf{(a)=11}\\\bullet\sf{(a+d)=11+9=20}$

For -9

$\bullet\sf{(a-d)=11-(-9)=11+9=20}\\\bullet\sf{(a)=11}\\\bullet\sf{(a+d)=11+(-9)=11-9=2}$

Answer 2

GIVEN:

Sum of 1st three terms of an AP = 33

Product of 1st and 3rd terms exceeds the 2nd term by 29.

TO FIND:

Arithmetic progression.

SOLUTION:

Let the first terms be (a - d), a and a + d

Sum of first three terms:

(a - d) + a + a + d = 33

==> 3a = 33

==> a = 33/3

==> a = 11

First term of AP = 11

Product of 1st and 3rd terms exceeds the 2nd term by 29.

(a - d) (a + d) = a + 29

a(a + d) - d(a + d) = a + 29

a² + ad - ad + d² = 11 + 29

11² + 11d - 11d + d² = 40

121 + d² = 40

==> d = √-81

==> d = ±9

Common Difference = ±9

Now, ap for +9:

a = 11 , a - d = 11 - 9 = 2 and a + d = 20

AP = 2, 11, 20......

ap for -9:

a = 11, a - d = a - (-d) = 11 + 9 = 20, a + d = 11 - 9 = 2

AP = 20, 11, 9.......

Therefore, the AP is 2, 11, 20... or 20, 11, 9....