Question

जर a=3आणि t6 =27असेल,तर त्या ankgnithi सरेचीच्या पहिल्या 6 पदांची बेरीज लिहा​

Answer 1

Given:

जर a = 3 आणि t₆ =27असेल

If a = 3 and t₆ = 27

To find:

तर त्या ankgnithi सरेचीच्या पहिल्या 6 पदांची बेरीज लिहा​

The sum of the first sixth term of the A.P.

Solution:

The formula of the nth term of an A.P. is as follows:

$\boxed{\bold{t_n = t + (n - 1)d}}$

where  

$t_n$ = last term, t = first term, n = no. of terms and d = common difference

We have,

$t_6 = 27$

∴ $t + (6 - 1)d = 27$

on substituting t = a = first term = 3

$\implies 3 + (6 - 1)d = 27$

$\implies 3 + 5d = 27$

$\implies 5d = 24$

$\implies d = \frac{24}{5}$ . . . . Equation 1

The formula of the sum of n terms of an A.P. is as follows:

$\boxed{\bold{S_n = \frac{n}{2}[2t + (n-1)d] }}$

where Sₙ = sum of n terms, t = first term, n = no. of terms and d = common difference

Now, using the formula above, we get

$S_6 = \frac{6}{2}[2(3) + (6-1)(\frac{24}{5}) ]$

$\implies S_6 = \frac{6}{2}[6 + (5)(\frac{24}{5}) ]$

$\implies S_6 = \frac{6}{2}[6 + 24 ]$

$\implies S_6 = \frac{6}{2}[30]$

$\implies S_6 = 3 \times 30$

$\implies \bold{S_6 = 90}$

Thus, the sum of the first sixth term of the A.P. is → 90.

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Answer 2

Given :- a = 3 , t(6) = 27 .

To Find :-

• S(6) = ?

Answer :-

we know that,

• T(n) = a + (n - 1)d .
• S(n) = (n/2)[2a + (n - 1)d] .

so,

→ T(6) = 27

→ a + (n - 1)d = 27

→ 3 + (6 - 1)d = 27

→ 3 + 5d = 27

→ 5d = 27 - 3

→ 5d = 24

→ d = (24/5) .

then,

→ S(n) = (n/2)[2a + (n - 1)d]

→ S(6) = (6/2)[2*3 + (6 - 1)(24/5)]

→ S(6) = 3[6 + 5(24/5)]

→ S(6) = 3[6 + 24]

→ S(6) = 3 * 30

→ S(6) = 90 (Ans.)

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