Question

How many terms of the AP: 9, 17, 25.
must be taken to a sum of 450
to get​

Answer 1

GiveN :

• AP is 9,17,25.......

To FinD :

• How many terms will be require to make sum as 450.

SolutioN :

We are given that AP is : 9,17,25......

• First term (a) = 9
• Common Difference (d) = 17 - 9 = 8
• Sum of terms (Sn) = 450

Use formula for Sum of AP

⇒Sn = n/2 [2a + (n - 1)d)]

⇒450 = n/2 [2(9) + (n - 1)8]

⇒450 * 2 = n[18 + 8n - 8]

⇒900 = n[8n + 10]

⇒900 = 8n² + 10n

⇒8n² + 10n - 900

Dividing by 2

⇒4n² + 5 - 450 = 0

Now, use method of splitting the middle term

⇒4n² - 40n + 45n - 450 = 0

⇒4n(n - 10) + 45(n - 10) = 0

⇒(n - 10)(4n + 45) = 0

⇒n - 10 = 0 & 4n + 45 = 0

⇒n = 10 & 4n = - 45

⇒n = 10 & n = -45/4

So, there are two values of n : 10 and - 45/4

As number of terms can't be negative. So, take positive values only.

⇒Number of terms = 10

Answer 2

${\sf{\color{magenta}{Correct~Question:-}}}$

How many terms of the AP: 9, 17, 25.

must be taken to get a sum of 450?

${\sf{\color{lawngreen}{Answer:-}}}$

a= 9

d = a2 - a1

d = 17 - 9

d = 8

Sn = 450

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

→450 = n/2 {2×9 + (n - 1)8}

→900 = n (18 + 8n - 8)

→900 = n (10 + 8n)

→900 = 10n + 8n²

→8n² + 10n - 900 = 0

→4n² + 5n - 450 = 0

→4n² + 45n - 40n - 450 = 0

→n (4n + 45) - 10 (4n + 45) = 0

→(n - 10)(4n + 45) = 0

→n = 10, -45/4

As the number of terms (n) cannot be negative.

Therefore, n = 10

Sum of first 10 terms of AP is 450.