Question

find the number of terms of 3+5+7......=620​

Answer 1

To solve this type of sum, we will use the concept of Arithmetic Progression.

▶️Working formula:-

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

So in the given series,

a = 3

d = 5 - 3 = 2

✍Applying the above formula, we get:-

➡️624 = n /2 { 6 + (n-1) 2}

➡️1248= 6n + 2n.n - 2n

➡️1248 - 4n - 2n.n =0

➡️2n.n + 4n -1248 = 0

➡️n.n + 2n - 624 = 0

➡️n.n + 26n - 24n -624 = 0

➡️n(n+26) - 24(n -26)=0

➡️n = -26, 24

But n cannot be negative so n = 24.

There are 24 terms in this AP.

Answer 2

$\huge{\bold{\boxed{\boxed{\mathtt{\red{ANSWER:-}}}}}}$

$\huge\text{\underline{\underline{n=24}}}$

$\huge{\bold{\boxed{\boxed{\mathtt{\red{EXPLANATION:-}}}}}}$

$\huge{\textsf{\pink{Given series is}}}$ $3 + 5 + 7 + \dots = 620$

Here We Have

first term (a) = 3

common difference (d) = 5-3 = 2

As it's in A.P. we can use the formula of sum

S = $\frac{n}{2}$ {2a + (n -1 ) d)}

We Have To find the no. of terms i.e. n

→According to the question,

S = $\mathbf{\frac{n}{2}}$ {2a + (n -1 ) d)}

⇒ 624 =n/2 { 6 + (n-1) 2}

⇒ 2 * 624 = n (6 + 2n- 2)

⇒ 1248 = 2n² + 4n

⇒ 1248 - 4n - 2n² = 0

⇒ 2n² + 4n -1248 = 0

⇒ n² + 2n - 624 = 0(∵ taking 2 common from each term)

⇒ n² + 26n - 24n - 624 = 0

⇒ n(n+26) - 24(n -26) = 0

⇒ (n-24) (n+26) = 0

⇒ (n-24) = 0 or (n+26) = 0

⇒ n= 24  or  n= -26

But n cannot be negative so n = 24.

$\huge{\textsf{\pink{\therefore The number of terms is 24.</p><p>}}}$