Ex 5.3 Q4 - How many terms of the AP: 9, 17, 25,... must be taken to give a sum of 636?

Ex 5.3 Q4 - How many terms of the AP: 9, 17, 25,... must be taken to give a sum of 636?

Solution:-

Let,

      There are n terms in the given AP.

*From given AP: 9, 17, 25...

     *First term = a = 9

     *Common difference = d = 8

     *Sum of n terms = Sₙ = 636

➙ We know that, sum of n terms of an AP is given by,

      Sₙ =

n / 2

[2a + (n - 1)d]

   ∴ 636 =

n / 2

[2(9) + (n - 1)8]

   ∴ 636 × 2 = n(18 + 8n - 8)

   ∴ 636 × 2 = n(10 + 8n)

   ∴ 636 × 2 = 10n + 8n²

   *Taking 2 as common

   ∴ 636 = 5n + 4n²

   ∴ 4n² + 5n - 636 = 0

   ∴ 4n² + 53n - 48n - 636 = 0

   ∴ n(4n + 53) - 12(4n + 53) = 0

   ∴ (4n + 53) (n - 12) = 0

Now,

      4n + 53 = 0           OR           n - 12 = 0

   ∴ 4n = -53                             ∴ n = 12

   ∴ n =

-53 / 4

But, n ≠ -53/4 as the number of terms can neither be negetive nor fractional,

   ∴ n = 12

Hence,

          12 terms of the AP: 9, 17, 25,... must be taken to give a sum of 636.

------------------------------------

Try This..

✒ The sum of first n terms of an AP is given by Sₙ = n/2 [2a + (n-1)d]. If a = 2, d = 3, and Sₙ = 90, find n.

✒ The sum of first n terms of an AP is 210. The first term is 7 and common difference is 3. Find n.

✒ How many terms of the AP: 5, 11, 17... are needed to make a sum of 450?

❌ Common Mistakes

Not identifying the correct values of a and d:

▸ Always double-check!

Errors in simplification:

▸ Be careful when expanding brackets or solving equations.

Forgetting to take the positive root:

▸ Since the number of terms can’t be negative, always pick the positive root.

Skipping the discriminant check:

▸ Ensure the square root value is a perfect square for ease of calculation.

📝 Related Questions:-

• Ex 5.3 Q6 - The first and the last terms of an AP are 17 and 350 respectively. If the common difference is 9, how many terms are there and what is their sum?
• Ex 5.3 Q15 - A contract on construction job specifies a penalty for delay of completion beyond a certain date as follows: ₹200 for the first day, ₹250 for the second day, ₹300 for the third day, etc., the penalty for each succeeding day being ₹50 more than for the preceding day. How much money the contractor has to pay as penalty, if he has delayed the work by 30 days?
• Ex 5.3 Q16 - A sum of ₹700 is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is ₹20 less than its preceding prize, find the value of each of the prizes.

Queries Solved:-

Class 10 Ex 5.3

Ex 5.3 Q4 Class 10

Class 10 Ex 5.3 Q4

Class 10 Chap 5 Ex 5.3 Q4

Class 10 Arithmetic Progression

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