Question

The seventh term of an AP is -4 and its 13th term is -16.
Find the (i)first term, (ii) common difference (iii) the nth term.
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Answer 1

Answer:

(i) First term = 8

(ii) Common difference = –2

(iii) nth term = 10 – 2n

Step-by-step explanation:

The nth term of an A.P is given by,

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

where

a denotes the first term

d denotes the common difference

_____________________

Given,

7th term = –4

a + (7 - 1)d = –4

a + 6d = –4

a = –4 – 6d

_____________________

It's also given, 13th term = –16

a + (13 – 1)d = –16

a + 12d = –16

–4 – 6d + 12d = –16

6d – 4 = –16

6d = –16 + 4

6d = –12

d = –12/6

d = –2

Therefore, common difference = –2

_____________________

First term :

a = –4 – 6d

a = –4 – 6(–2)

a = –4 + 12

a = 8

First term = 8

_____________________

nth term :

= a + (n – 1)d

= 8 + (n – 1)(–2)

= 8 – 2n + 2

= 10 – 2n

Answer 2

$\bf\huge \pink{ANSWER}$

$\small \text \green{Let's assume that the first term be a }$

$\small \text \green{and common difference be b }$

$\bf\huge\pink{Given}$

$\small \text \blue{The 7 th term of A. P = -4 }$

$\bf{ = a + 6d = - 4} \: \: (1)$

$\small\text\blue{The 13th term of \: A.P is -16. }$

$\bf{ = a + 12d = - 16} \: \: (2)$

$\small \text \pink{From equation (1) and (2)}$

$\sf{ \cancel{ a }+ 6d = - 4} \: \: (1)$

$\sf{ \cancel{ a }+ 12d = - 16} \: \: (2)$

_-____-______+__

$\bf \boxed{{ \purple{ - 6d = + 12}}}$

$\bf \huge \boxed{d = - 2}$

$\bf\huge\pink{Therefore}$

$\sf{ = a + 6d = - 4}$

$\sf > > {a = - 4 - 6 \times ( - 2)}$

$\bf{ > > a = ( - 4 )+ 12}$

$\bf \huge \boxed{a= 8}$

$\bf\huge \pink{Solution}$

$\small \text{(i)first term} \ \: \bf \small\boxed{a= 8}$

$\small \text{(ii) common difference} \: \bf \small \boxed{d = - 2}$

(iii) n th term :

= a + (n – 1)d

= 8 + (n – 1)(–2)

= 8 – 2n + 2

= $\bf \small\boxed{10-2n}$

$\bf\huge \pink{Thanks}$