Question

If sum of 3rd and 8th terms of an A.P. is 7 and sum of 7th and 14th terms is -3 then
Find the 10th term.​

Answer 1

EXPLANATION.

Sum of 3rd and 8th terms of an A.P. = 7.

Sum of 7th and 14th terms of an A.P. = -3.

As we know that,

General term of an A.P.

⇒ Tₙ = a + (n - 1)d.

Using this formula in equation, we get.

⇒ T₃ + T₈ = 7. - - - - - (1).

⇒ a + 2d + a + 7d = 7.

⇒ 2a + 9d = 7. - - - - - (1).

⇒ T₇ + T₁₄ = -3. - - - - - (2).

⇒ a + 6d + a + 13d = -3.

2a + 19d = -3. - - - - - (2).

From equation (1) & (2), we get.

⇒ -10d = 10.

⇒ d = -1.

Put the value of d = -1 in equation (1), we get.

⇒ 2a + 9(-1) = 7.

⇒ 2a - 9 = 7.

⇒ 2a = 7 + 9.

⇒ 2a = 16.

⇒ a = 8.

To find : 10th term of an A.P.

⇒ T₁₀. = a + (10 - 1)d.

⇒ T₁₀ = a + 9d.

Put the value in the equation, we get.

⇒ T₁₀ = 8 + 9(-1).

⇒ T₁₀ = 8 - 9.

⇒ T₁₀ = -1.

                                                                                                                       

MORE INFORMATION.

Supposition of an A.P.

(1) = Three terms as : a - d, a, a + d.

(2) = Four terms as : a - 3d, a - d, a + d, a + 3d.

(3) = Five terms as : a - 2d, a - d, a, a + d, a + 2d.

Answer 2

$\huge\bf\underline\mathfrak{Answer :}$

• $\text{10th term}$ = $\sf\red{-1}$

$\huge\bf\underline\mathfrak{Step \: by \: step \: explanation :}$

$\huge\bf\underline\mathfrak{Given :}$

• $\text{3rd term + 8th term of the A.P. = 7}$
• $\text{7th term + 14th term = -3}$

$\huge\bf\underline\mathfrak{To \: find :}$

• $\text{10th term of the A.P.}$

$\huge\bf\underline\mathfrak{Solution :}$

$\sf\underbrace{General \: term \: of \: an \: A.P. \: :-}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀$\sf a_n=a \: + \: ( \: n - 1 \: )d$

Where,

• a = $\text{First term}$
• d = $\text{Common difference}$

$\underline\text{From the above formula, we can say :-}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀$\text{3rd term}$ = $\sf\purple{a+2d}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀$\text{8th term}$ = $\sf\purple{a+7d}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀$\text{7th term}$ = $\sf\purple{a+6d}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀$\text{14th term}$ = $\sf\purple{a+13d}$

$\sf\underbrace{According \: to \: question \: :-}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀$\color{green}\star\:\tt{Case \: 1 \: :-}$

$\sf{a_3+a_8 = 7}$

$\implies$ $\sf{a + 2d + a + 7d = 8 }$

$\implies$ $\sf{2a + 9d = 8}$ --- $\text{Equation (i)}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀$\color{green}\star\:\tt{Case \: 2 \: :-}$

$\sf{7th \: term + 14th \: term}$ = -3

$\implies$ $\sf{a + 6d + a + 13d = -3 }$

$\implies$ $\sf{2a + 19d = -3}$ --- $\text{Equation (ii)}$

$\sf\underbrace{Solving \: both \: equations \: :-}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀$\sf\red{a = 8}$ and $\sf\red{d = -1}$

$\sf\underbrace{Finding \: 10th \: term \: :-}$

$\sf{10th \: term = a + 9d }$

$\implies$ $\sf{8 + 9(-1)}$

$\implies$ $\sf\purple{-1}$