Question

If in a Arithmetic progression

$\sf d=2,S_{15}=285$

Find,

$\sf a$​

Answer 1

Question : -

In a Arithmetic Progression,

d = 2, S₁₅ = 285

Find the value of a ?

ANSWER

Given : -

d = 2, S₁₅ = 285

Required to find : -

• value of a ?

Solution : -

Given that;

d = 2, d = 2, S₁₅ = 285 = 285

Since, we know that

S_(nth) = (n)/(2) [2a+(n-1)d]

Where,

• n = number of terms
• a = first term
• d = common difference

Now,

Here S_(nth) indicates S₁₅

So,

• n = 15
• d = 2
• S₁₅ = 285

Substituting these values in the formula,

_______________________

S₁₅ = (15)/(2) [2a+(15-1)2]

285 = (15)/(2) [2a+(14)2]

285 = (15)/(2) [2a+28]

(285)/(15) = (1)/(2) [2a+28]

19 = (1)/(2) 2[a+14]

19 = (2[a+14])/(2)

19 = a+14

19-14 = a

5=a

» a = 5

Hence,

• Value of a = 5

Answer 2

$\huge\purple{\mid{\underline{\overline{\texttt{Question}}}\mid}}$

If in a Arithmetic Progression, the common difference is 2 and the sum of first fifteen terms is 285. Then find the first term of the arithmetic progression??

$\huge\pink{\mid{\fbox{\tt{Answer↴}}\mid}}$

First term of the A.P (a) = 5

$\huge\mathtt{\fcolorbox{pink}{cyan}{\purple{Solution}}}$

GIVEN -

Common Difference(d) = 2

no. of terms(n) = 15

Sum of First 15 terms $\tt( S_{15}) = 285$

TO FIND -

The first term of the arithmetic progression(a)

FORMULA USED -

$\tt\implies (S_{ {n}^{th} }) = \frac{n}{2}[2a + (n - 1)d]$

where,

n = no. of terms

a = first term of that A.P

d = common difference

$\tt \: S _{ {n}^{th} } = sum \: of \: {n}^{th } term \: of \: the \: A.P$

SOLUTION -

$\tt\implies (S_{ {n}^{th} }) = \frac{n}{2}[2a + (n - 1)d]$

Putting all the values,

$\tt\implies (S_{ {15}^{th} }) = \frac{15}{2}[2a + (15 - 1)2]$

$\tt\implies 285 = \frac{15}{2}[2a + (14)2]$

$\tt\implies \frac{285 \times 2}{15} = [2a + (14)2]$

$\tt\implies \frac{285 \times 2}{15} = (2a + 28)$

$\tt\implies {19 \times 2} = 2a + 28$

$\tt\implies 38 = 2a + 28$

$\tt\implies 38 - 28 = 2a$

$\tt \implies10 = 2a$

$\tt \implies5 = a$

Hence,

The first term of the arithmetic progression is 5.