Question

GENIUS PLZ DO THIS ...........​

Answer 1

Answer:

2, 10, 18...

Step-by-step explanation:

$\underline{The \:sum \:of \:first \:seven\: terms\: of\: an}$ $\underline{arithmetic \:progression\: is \:182.}$

Means $S_{7}$ = 182.

$\underline{If \:it's\: 4th\: and \:7th \:terms \:are\: in \:the \:ratio }$ $\underline{1:5.}$

Means..

$\dfrac{a\:+\:3d}{a\:+\:16d}$ = $\dfrac{1}{5}$

$\underline{Then\: find\: the \:arithmetic\: progression.}$

» $a_{n}$ = a + (n - 1)d

$a_{4}$ = a + (4 - 1)d

= a + 3d

$a_{17}$ = a + (17 - 1)d

= a + 16d

$\dfrac{a\:+\:3d}{a\:+\:16d}$ = $\dfrac{1}{5}$

Now cross multiply them..

5a + 15d = a + 16d

5a - a = 16d - 15d

$\textbf{4a = d}$ ...(1)

» $S_{n}$ = $\dfrac{n}{2}$ (a + $a_{n}$)

$S_{7}$ = $\dfrac{7}{2}$ [a + a + (n - 1)d]

182 = $\dfrac{7}{2}$ [(2a + (7 - 1)d]

182 = $\dfrac{7}{2}$ (2a + 6d)

182 = $\dfrac{7}{2}$ [2a + 6 (4a)]

{From (1)}

182 = $\dfrac{7}{2}$ (2a + 24a)

182 = $\dfrac{7}{2}$ × 2 (a + 12d)

182 = 7 × 13a

$\dfrac{182}{7}$ = 13a

13a = 26

a = $\dfrac{26}{13}$

$\textbf{a = 2}$

Put value of a in (1)

4 (2) = d

$\textbf{d = 8}$

A.P (Arithmetic Progression)

a, a + d, a + 2d...

2, 2 + 8, 2 + 2(8)...

$\textbf{2, 10, 18...}$

Answer 2

Question : The sum of seven terms of an Arithmetic progression is 182. If it's 4th and 17th terms are in the ratio 1:5, find the arithmetic progression.

Step by step explanation :

a = First term

d = common difference

T4 = a + 3d

T17 = a + 16d

As per your question,

a + 3d/ a + 16d = 1/5

5a + 15d = a + 16d

4a = d....(1)

Now,

S7 = 182

.°. 182 = 7/2 [ 2a + (7-1) d ]

182 = 7 × (a + 3d)

26 = a + 3d....(2)

From (1),

26 = a + 3 ( 4a )

26 = 13a

a = 2

From (1),

4 (2) = d

8 = d

Therefore,

t1 = 2

t2 = a + d = 2 + 8 = 10

t3 = t2 + d = 10 + 8 = 18.

Thus,

The A.P formed will be 2,10,18...

Hope it helps!