Question

Maths Question : Ap!


- If the sum of n terms of the AP 25,22,19...... Is 116 find l that is last term .


Give proper solution

Answer 1

Heya !
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★Arithmetic Progressions ★
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→ Given A.P = 25 , 22 , 19.........


• First term a = 25

• Common Difference d = 22 - 25 = -3

• Sn = 116


→last term l = ?


Now we have the formula for the sum of n terms ,

=> Sn = n/2 [ 2a + ( n - 1 ) d ]


$= > 116 = \frac{n}{2} (2 \times 25 + (n - 1) - 3) \\ \\ = > 232 = n(50 - 3n+ 3)$

=> We have the Quadratic Equation :


→ 3n² - 53n - 232 = 0


Now I am solving it further using the Quadratic formula . You can opt any method.
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•Here a = 3 , b = -53 and c= -232

We have d = b² - 4ac

=> d = (53)² - 4 × 3 × 232

=> d = 2809 - 2784


•°• d = 25


Then we have x = - b +- √d/2a


We will get two values here:


=) x = 53 + 5 / 6 ( neglect this value as it comes in fractions )

=) x = 53 - 5/6 = 48/6 = 8 ✔



°•° n = 8
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=> Now we have to find the last term . There are two ways :


→First method= : Sn = n/2 ( a + l )


→ 116 = 8/2 ( 25 + l )

=> 116 = 100 + 4l

=> 4l = 16


•°• Last term = 4 ✔


→Second Method : an = a + ( n - 1 )d

=> an = 25 + 7 × -3

=> an = 25 - 21


•°• Last term = 4 ✔


★The last term of the A.P is 4
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Answer 2

Holla!


AP - 25, 22 , 19


a = 25 and d = -3 and Sn = 116


Use the formula of Sn


Sn = n/2 ( 2a+(n-1)d)


116 = n/2 ( 50 - 3n +3)


= 232 = n ( 53-3n)

= 3n² - 53n - 232 = 0


Solve the equation we get n=8 ♥


Now last term an = a + (n-1)d

= 25 + 7d


= 25 - 21 = 4