Question

The sum of the first and 19th terms of an arithmetic sequence is 60.The sum of it's first and 20th terms is 62. a) what is it's comma difference. b) what is the sum of it's 8th and 12th terms. c) Find it's 10th term. ​

Answer 1

EXPLANATION.

Sum of the first and 19th term of an AP = 60.

sum of first and 20th term of an AP = 62.

Formula of nth term of an AP.

⇒ An = a + ( n - 1 ) d.

⇒ a + a + 18d = 60

⇒ 2a + 18d = 60.

⇒ a + 9d = 30  .........(1)

⇒ a + a + 19d = 62

⇒ 2a + 19d = 62.   ........(2).

from equation (1) and (2) we get,

divide equation (1) by 2

divide equation (2) by 1 we get,

2a + 18d = 60

2a + 19d = 62

-     -          -

               

- d = = -2

d = 2

put the value of d = 2 in equation (1) we get,

⇒ a + 19(2) = 60.

⇒ a + 18 = 60.

⇒ a = 42.

First term = a = 42.

(a) = common difference =d = 2.

(b) = sum of its 8th and 12th term of nan AP.

⇒ a + 7d + a + 11d

⇒ 2a + 18d.

⇒ 2(42) + 18(2).

⇒ 84 + 36

⇒ 120.

(c) = its 10th term.

⇒ a + 9d.

⇒ 42 + 9(2).

⇒ 42 + 18 = 60.

Answer 2

Answer:

$\huge{\fcolorbox{black} {black}{\color{yellow}{ANSWER}}}$

$\pink{GIVEN : }$

Sum of first and 19th term = 60

Sum of first and 20th term = 62

$\pink{SOLUTION : }$

Formula for nth term of any AP =

$\blue{\boxed{\green{A(n) = a + (n - 1)d}}}$

where :

• a is first term of an AP
• n is number of terms
• d is the common difference where d = A(n) — A(n-1)

$\implies a + a + 18d = 60$

$\implies 2a + 18d = 60$

Dividing the equation by 2

$\implies a + 9d = 30..........(1)$

_______________

$\implies a + a + 19d = 62$

$\implies 2a + 19d = 62..........(2)$

_______________

Now we will solve 1 and 2 equations by elimination by substitution

For this we multiply (1) by 2

$\implies 2a + 18d = 60..........(3)$

Subtracting (3) from (2)

$</strong><strong>\</strong><strong>implies</strong><strong> </strong><strong>2a + 19d - 2a - 18d = 62 - 60$

$</strong><strong>\</strong><strong>boxed</strong><strong>{</strong><strong>\</strong><strong>longrightarrow</strong><strong> </strong><strong>d = 2</strong><strong>}</strong><strong>$

_____________

Substituting d in (1)

$\implies a + 18 = 30$

$\implies a = 12$

_____________

A(8) = a + 7d

= 12 + 14

= 26

A(12) = a + 11d

= 12 + 22

= 34

[tex]\boxed{SUM = 60}[/tex]

_____________

A(10) = a + 9d

= 12 + 18

= 30

_____________

KNOWLEDGE EXPRESS :

AP is a series which have a common difference between any 2 consecutive terms.

Formula for sum of nth term =

$</strong><strong>Sum</strong><strong> = \frac{1}{2} \times ( 2a + (n - 1)d)$

OR

$Sum = \frac{1}{2} \times (a + l)$

Where

• a is first term and
• l is last term or the nth term of the AP

______________

Hope it helps

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