The sum of the third and ...

The sum of the third and the seventh terms of an AP is 6 and their product is 8.find the sum of first sixteen terms of the AP
  • 1 answers

Sia 🤖 8 months ago

Let the first term and the common difference of the AP be a and d respectively.
According to the question,
Third term + seventh term = 6
{tex} \Rightarrow {/tex} [a + (3 - 1)d] + [a + (7 - 1)d] = 6 = a + (n - 1)d
{tex} \Rightarrow {/tex} (a + 2d) + (a + 6d) = 6 {tex} \Rightarrow {/tex} 2a + 8d = 6
{tex} \Rightarrow {/tex} a + 4d = 3 ..... (1)
Dividing throughout by 2 &
(third term) (seventh term) = 8
{tex} \Rightarrow {/tex} (a + 2d) (a + 6d) = 8
{tex} \Rightarrow {/tex} (a + 4d - 2d) (a + 4d + 2d) = 8
{tex} \Rightarrow {/tex} (3 - 2d) (3 + 2d) = 8
{tex} \Rightarrow {/tex} 9 - 4d2 = 8
{tex} \Rightarrow 4{d^2} = 1 \Rightarrow {d^2} = \frac{1}{4} \Rightarrow d + \pm \frac{1}{2}{/tex}
Case I, when {tex}d = \frac{1}{2}{/tex}
Then from (1), {tex}a + 4\left( {\frac{1}{2}} \right) = 3{/tex}
{tex} \Rightarrow {/tex} a + 2 = 3 {tex} \Rightarrow {/tex} a = 3 - 2 {tex} \Rightarrow {/tex} a = 1
{tex}\therefore {/tex} Sum of first sixteen terms of the AP = S16
{tex} = \frac{{16}}{2}[2a + (16 - 1)d]{/tex} {tex}\because {S_n} = \frac{n}{2}[2a + (n - 1)d]{/tex}
= 8[2a + 15d]
{tex} = 8[2(1) + 15(\frac{1}{2})]{/tex}
{tex} = 8[12 + \frac{{15}}{2}]{/tex}
{tex} = 8[\frac{{19}}{2}]{/tex}
{tex} = 4 \times 19 = 76{/tex}
Case II. When {tex}d = - \frac{1}{2}{/tex}
Then from (1),
{tex}a + 4\left( { - \frac{1}{2}} \right) = 3{/tex}
{tex} \Rightarrow {/tex} a - 2 = 3 {tex} \Rightarrow {/tex} a = 3 + 2 {tex} \Rightarrow {/tex} a = 5
{tex}\therefore {/tex} Sum of first sixteen terms of the AP = S16
{tex} = \frac{{16}}{2}[2a + (16 - 1)d]{/tex} {tex}\because {S_n} = \frac{n}{2}[2a + (n - 1)d]{/tex}
{tex} = 8[2a + 15d] = 8\left[ {2(5) + 15\left( { - \frac{1}{2}} \right)} \right] = 8\left[ {10 - \frac{{15}}{2}} \right] = 8\left[ {\frac{5}{2}} \right] = 20{/tex}

Related Questions:

Buy Complete Study Pack

Subscribe complete study pack and get unlimited access to selected subjects. Effective cost is only ₹ 12.5/- per subject per month. Based on the latest CBSE & NCERT syllabus.

myCBSEguide App

myCBSEguide

Trusted by 70 Lakh Students

CBSE Test Generator

Create papers in minutes

Print with your name & Logo

Download as PDF

3 Lakhs+ Questions

Solutions Included

Based on CBSE Blueprint

Best fit for Schools & Tutors

Work from Home

  • Work from home with us
  • Create questions or review them from home

No software required, no contract to sign. Simply apply as teacher, take eligibility test and start working with us. Required desktop or laptop with internet connection