Question

How many terms of AP : 9, 17, 25....must be taken to give a sum of 636??​

Answer 1

Answer: n = 12

Given AP is 9, 17, 25,…

We know that,

Sn = n/2[2a + (n − 1)d]

Here we have,

The first term (a) = 9

Sum of n terms (Sn) = 636

Common difference of the A.P. (d) = a2 – a1

d = 17 – 9

d = 8

Substituting the values in Sn, we get

636 = n/2[2(9) + (n − 1)(8)]

636 = n/2[18 + (8n − 8)]

636 × 2 = n× [10 + 8n]

1271 = 10n + 8n2

Now, we get the following quadratic equation,

8n2 + 10n – 1272 = 0

4n2+ 5n – 636 = 0

On solving by factorization method, we have

4n2 – 48n + 53n – 636 = 0

4n(n – 12) + 53(n – 12) = 0

(4n + 53)(n – 12) = 0

Either 4n + 53 = 0

n = -53/4

Or, n – 12 = 0

n = 12

Since, the number of terms cannot be a fraction.

∴ The number of terms (n) = 12

Answer 2

$\large\underline{\sf{Solution-}}$

Given AP series is

$\rm :\longmapsto\:9, \: 17, \: 25, \: - - -$

So it implies,

First term of AP, a = 9

Common difference of AP, d = 17 - 9 = 8

Now, it is further given that

Sum of terms of AP = 636

Let assume that number of terms required be n to get the sum of AP series is 636.

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ Sum of n  terms of an arithmetic sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{S_n\:=\dfrac{n}{2} \bigg(2 \:a\:+\:(n\:-\:1)\:d \bigg)}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

• Sₙ is the sum of n terms of AP.

• a is the first term of the sequence.

• n is the no. of terms.

• d is the common difference.

Now, on substituting the values in above formula, we get

$\rm :\longmapsto\:636 = \dfrac{n}{2} \bigg(2 \times 9 + (n - 1)8\bigg)$

$\rm :\longmapsto\:636 = \dfrac{n}{2} \bigg(18 + 8n -8\bigg)$

$\rm :\longmapsto\:636 = \dfrac{n}{2} \bigg(10 + 8n\bigg)$

$\rm :\longmapsto\:636 = \dfrac{n}{2} \times 2 \times \bigg(5 + 4n\bigg)$

$\rm :\longmapsto\:636 = n \bigg(5 + 4n\bigg)$

$\rm :\longmapsto\: {4n}^{2} + 5n = 636$

$\rm :\longmapsto\: {4n}^{2} + 5n - 636 = 0$

$\rm :\longmapsto\: {4n}^{2} - 48n+ 53n - 636 = 0$

$\rm :\longmapsto\:4n(n - 12) + 53(n - 12) = 0$

$\rm :\longmapsto\:(n - 12)(4n + 53) = 0$

$\rm :\implies\:n = 12 \: \: \: or \: \: \: n = - \dfrac{53}{4} \: \{rejected \}$

Hence,

$\: \: \underbrace{\boxed{ \qquad \bf{ Number \: of \: terms, \: n\: = \: 12 \qquad}}}$

Additional Information :-

↝ nᵗʰ term of an arithmetic sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{a_n\:=\:a\:+\:(n\:-\:1)\:d}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

• aₙ is the nᵗʰ term.

• a is the first term of the sequence.

• n is the no. of terms.

• d is the common difference