The sum of third, fourth and fifth terms of an arithmetic series is 99. If the first term is 3, then find the value of fifth term.

 The sum of third, fourth and fifth terms of an arithmetic series is 99. If the first term is 3, then find the value of fifth term.

Ans:- 

first term of the arithmetic series as ${𝑎}_1=3$ and the common difference as $𝑑$.

The formula for the nth term of an arithmetic series is given by: ${𝑎}_{𝑛}={𝑎}_1+(𝑛-1)𝑑$

We are given that the sum of the third, fourth, and fifth terms of the series is 99. We can express this as: ${𝑎}_3+{𝑎}_4+{𝑎}_5=99$

Substitute the expressions for ${𝑎}_3,{𝑎}_4,$ and ${𝑎}_5$ using the formula for the nth term: $({𝑎}_1+2𝑑)+({𝑎}_1+3𝑑)+({𝑎}_1+4𝑑)=99$ $(3+2𝑑)+(3+3𝑑)+(3+4𝑑)=99$

 Substitute the expressions for

${𝑎}_3,{𝑎}_4,$ and ${𝑎}_5$ using the formula for the nth term: $({𝑎}_1+2𝑑)+({𝑎}_1+3𝑑)+({𝑎}_1+4𝑑)=99$ $(3+2𝑑)+(3+3𝑑)+(3+4𝑑)=99$

Combine like terms and solve for $𝑑$: $9+9𝑑=99$ $9𝑑=90$ $𝑑=10$

Now that we have found the common difference $𝑑=10$, we can find the fifth term ${𝑎}_5$ using the formula for the nth term: ${𝑎}_5={𝑎}_1+(5-1)𝑑$

  

a5​=3+(4)(10) ${𝑎}_5=3+40$ ${𝑎}_5=43$

Therefore, the value of the fifth term in the arithmetic series is 43.