Question

Hélló gúys , plz solve it's urgent .
#MATHS TIME .

The sum of first 6 terms of an AP is 42 . The ratio of its 10th term to 30th term is 1 : 3 . Find the first and the 13th term of the AP .


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Answer 1

The sum of first 6 terms of an AP is 42 . The ratio of its 10th term to 30th term is 1 : 3 . Find the first and the 13th term of the AP .

Good question,

Here is your answer!

Let the first term be a and common difference be d.

S6 = 42

=) 6/2 * {2a + (6-1)d} = 42

=) 3(2a + 5d) = 42

=) 2a + 5d = 42/3

=) 2a + 5d = 14 - - 1)

Given,

=) a10/a30 = 1/3

=) (a +9d)/(a + 29d) = 1/3

=) 3(a+9d) = a+29d

=) 3a + 27d + a + 29d

=) 3a - a = 29d - 27d

=) 2a = 2d

=) a = d, - - 2)

Put d = a in eq 1).

=) 2a + 5a = 14

=) 7a = 14

=) a = 14/7

=) a = 2 = d

Hence first term = 2,

a13 (13th term) = a + (13-1)d

= a + 12d

= 2 + 12*2

= 2 + 24

= 26.

Answer 2

▶ Question :-

→ The sum of first 6 terms of an AP is 42 . The ratio of its 10th term to 30th term is 1 : 3 . Find the first and the 13th term of the AP .



▶ Answer :-

→ The first term is 2 and 13th term term is 26 .



▶ Step-by-step explanation :-


$\huge \pink{ \mid{ \underline{ \overline{ \tt Solution :- }} \mid}}$


→ Let a be the first term and d be the common difference of the given AP . Then,


→ $\sf a_{10} = a + 9d \: \: and \: \: a_{30} = a + 29d .$ .


$\sf \therefore \frac{a_{10}}{a_{30}} = \frac{1}{3} . \\ \\ \sf \implies \frac{a + 9d}{a + 29d} = \frac{1}{3} . \\ \\ \sf \implies3a + 27d = a + 29d. \\ \\ \sf \implies \cancel2a = \cancel2d. \\ \\ \sf \large \implies a = d. \\ \\ \\ \sf Also, S_n = \frac{n}{2} \bigg(2a + (n - 1)d \bigg). \\ \\ \sf \implies S_6 = \frac{6}{2} (2a + 5d). \\ \\ \sf = 3(2a + 5d). \\ \\ \sf = (6a + 15d). \\ \\ \sf = (6a + 15a). \: \: \: \: \{ \because d = a \} \\ \\ \large \sf = 21a.$


$\sf But, S_6 = 42 . ( given ) .\\ \\ \sf \therefore 21a = 42 . \\ \\ \huge \orange{ \boxed{ \sf \implies a = 2.}}$


Thus, a = 2 and d = 2 .


•°• 13th term , $a_13$ = ( a + 12d ) .

= ( 2 + 12 × 2 ) .

$\huge \blue { \boxed{ \sf = 26 . }}$



✔✔ Hence, the first term is 2 and 13th term term is 26 ✅✅ .



$\huge \red{ \boxed{ \boxed{ \mathscr{THANKS}}}}$