Question

the sum of the third and seventh term of an AP is 40 and the sum sixth and 14th terms is 70 .Find the sum of first ten of the AP
​

Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

FORMULA TO BE IMPLEMENTED

If in an arithmetic progression

First term = a

Common Difference = d

Then

$\sf{1. \: \: n \: th \: \: term = a + (n-1)d}$

$\displaystyle \: \sf{2. \:Sum \: of \: the \: first \: n \: terms \: = \frac{n}{2} \{2a + (n - 1)d \}\: \: }$

GIVEN

• The sum of the third and seventh term of an AP is 40
• The sum sixth and 14th terms is 70

TO DETERMINE

The sum of first ten of the AP

sum of first ten of the AP

CALCULATION

Let first term = a

Common Difference = d

$\sf{ \: 3 rd \: term \: = a + 2d \: }$

$\sf{ \: 7th \: term \: = a + 6d \: }$

$\sf{ \: 6th \: term \: = a + 5d \: }$

$\sf{ \: 14th \: term \: = a + 13d \: }$

By the given condition

$\sf{ a + 2d + a + 6d = 40\: \: }$

$\implies \: \sf{ 2a + 8d = 40\: \: }$

$\implies \: \sf{ a + 4d = 20\: \: }........(1)$

Again

$\sf{ a + 5d + a + 13d = 70\: \: }$

$\implies \: \sf{ 2a + 18d = 70\: \: }$

$\implies \: \sf{ a + 9d =35\: \: }.......(2)$

Equation (2) - Equation (1) gives

$\sf{ 5d = 15\: \: }$

$\implies \: \sf{ d = 3\: \: }$

From Equation (1)

$\implies \: \sf{ a = 20 - 4d\: \: }$

$\implies \: \sf{ a = 20 - 12 = 8\: \: }$

Hence the sum of first ten terms

$\displaystyle \: \sf{= \frac{10}{2} \times \{2a + (10 - 1)d \}\: \: }$

$\displaystyle \: \sf{= 5 \times (2a + 9d) \: }$

$\displaystyle \: \sf{= 5 \times (a +a + 9d) \: }$

$\displaystyle \: \sf{= 5 \times (8 +35) \: }$

$\displaystyle \: \sf{= 5 \times 43 \: }$

$\displaystyle \: \sf{=215 \: }$

━━━━━━━━━━━━━━━━

LEARN MORE FROM BRAINLY

Find the 100th term of an AP whose nth term is 3n+1

https://brainly.in/question/22293445