Question

if the 8term of an ap is 37 and 15term is 15 more than the 12 term find the sum of the first 20 term of the ap ​

Answer 1

Answer:

The sum of first 20 terms is 990.

Step-by-step explanation:

Given :-

• The 8th term of an AP is 37 and 15th term is 15 more than the 12th term.

To find :-

• Sum of the first 20 terms of the AP.

Solution :-

Formula used :

${\boxed{\sf{T_n=a+(n-1)d}}}$

• a = first term.
• d = common difference.

★ The 8th term of the AP is 37.

$\to\sf{T_8=a+(8-1)d}$

$\to\sf{37=a+7d}$

$\to\sf{a+7d=37..................(i)}$

Now find the 15th term and 12th term of the AP.

★ $\sf{T_{15}=a+(15-1)d}$

$\to\sf{T_{15}=a+14d}$

★ $\sf{T_{12}=a+(12-1)d}$

$\to\sf{T_{12}=a+11d}$

According to the question ,

$\to\sf{T_{15}=T_{12}+15}$

$\to\sf{a+14d=a+11d+15}$

$\to\sf{14d-11d=15}$

$\to\sf{3d=15}$

→ d = 5

Now put d = 5 in eq(i).

a+7d = 37

→ a + 7×5 = 37

→ a + 35 = 37

→ a = 37-35

→ a = 2

Formula used :

${\boxed{\sf{S_n=\dfrac{n}{2}(2a+(n−1)d)}}}$

Sum of first 20 terms ,

=20/2[2×2+(20−1)×5]

=10[4+19×5]

=10[4+95]

=10×99

=990

Therefore, the sum of first 20 terms is 990.

Answer 2

Solution :

$\underline{\bf{Given\::}}$

If the 8th term of an A.P. is 37 & 15th term is 15 more than the 12 term.

$\underline{\bf{Explanation\::}}$

We know that formula of an A.P;

$\boxed{\bf{a_n = a+(n-1)d}}}$

• a is the first term.
• d is the common difference.
• n is the term of an A.P.

A/q

→ a8 = 37

→ a + (8-1)d = 37

→ a + 7d = 37

→ a = 37 - 7d...................(1)

&

→ a15 = a12 + 15

→ a + (15-1)d = a + (12-1)d + 15

→ a + 14d = a + 11d + 15

→ 37 - 7d + 14d = 37 - 7d + 11d + 15      [from (1)]

→ 37 + 7d = 37 + 4d + 15

→ 37 + 7d = 52 + 4d

→ 7d - 4d = 52 - 37

→ 3d = 15

→ d = 15/3

→ d = 5

∴ Putting the value of d in equation (1),we get;

→ a = 37 - 7(5)  

→ a = 37 - 35

→ a = 2

Now;

As we know that formula of the sum of an A.P;

$\boxed{\bf{S_n = \frac{n}{2} \bigg[2a+(n-1)d\bigg]}}}$

→ S20 = 20/2 [2(2) + (20-1) (5)

→ S20 = 10 [4 + 19 × 5]

→ S20 = 10 [4 + 95]

→ S20 = 10 × 99

→ S20 = 990

Thus;

The sum of the first 20 term of the A.P. will be 990 .