The number of terms in an A.P. is even. The s | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let the total number of terms be $2n$,the first term be $a$,and the common difference be $d$.The terms are $a, a+d, a+2d, ..., a+(2n-1)d$.The odd-

Vedclass · Accepted Answer

$8$

Vedclass · Answer

(B) Let the total number of terms be $2n$,the first term be $a$,and the common difference be $d$.The terms are $a, a+d, a+2d, ..., a+(2n-1)d$.The odd-positioned terms are $a, a+2d, ..., a+(2n-2)d$. There are $n$ such terms.Sum of odd terms $S_o = \frac{n}{2}[2a + (n-1)(2d)] = n[a + (n-1)d] = 24$  --- $(i)$The even-positioned terms are $a+d, a+3d, ..., a+(2n-1)d$. There are $n$ such terms.Sum of even terms $S_e = \frac{n}{2}[2(a+d) + (n-1)(2d)] = n[a+d+(n-1)d] = 30$  --- $(ii)$Subtracting $(i)$ from $(ii)$:$n(a+d+(n-1)d) - n(a+(n-1)d) = 30 - 24$$nd = 6$  --- $(iii)$Given the last term exceeds the first term by $10\frac{1}{2} = \frac{21}{2}$:$(a+(2n-1)d) - a = \frac{21}{2}$$(2n-1)d = \frac{21}{2}$$2nd - d = \frac{21}{2}$Substitute $nd = 6$ into the equation:$2(6) - d = \frac{21}{2}$$12 - d = 10.5$$d = 1.5 = \frac{3}{2}$Using $nd = 6$:$n(\frac{3}{2}) = 6 \Rightarrow n = 4$Total number of terms $= 2n = 2 	imes 4 = 8$.

The number of terms in an $A.P.$ is even. The sum of the odd terms is $24$ and the sum of the even terms is $30$. If the last term exceeds the first term by $10\frac{1}{2}$,then the number of terms in the $A.P.$ is:

Similar Questions

Consider an $A.P.$ of positive integers,whose sum of the first three terms is $54$ and the sum of the first twenty terms lies between $1600$ and $1800$. Then its $11^{\text{th}}$ term is:

If $S_1, S_2, S_3, \dots, S_m$ are the sums of $n$ terms of $m$ $A.P.s$ whose first terms are $1, 2, 3, \dots, m$ and common differences are $1, 3, 5, \dots, 2m - 1$ respectively,then $S_1 + S_2 + S_3 + \dots + S_m = $

Which of the following sequences is an arithmetic sequence?

If the first,second and last terms of an $A.P.$ are $a, b$ and $2a$ respectively,then its sum is:

If the sum of the first $n$ terms of an arithmetic progression is $cn^2$,what is the sum of the squares of these $n$ terms?

Vedclass Test Series

Exam Paper Generator

Online Exam Module