Three numbers are in AP Another 3 numbers are in GP The sum of first term of the - Maths - Sequences and Series

Question

Three numbers are in AP Another 3 numbers are in GP The sum of
first term of the AP and
the first term of the GP is 85, the sum of second term of AP and
the second term of the GP is
76 and that of the 3rd term of AP and 3rd term of GP is 84 The sum
of the AP is 126 Find each
term of AP and GP

Shivam Mishra · Accepted Answer

Dear Student,
Please find below the solution to the asked query:

Let terms of A P
 be a-d,a,a+d and that of G P
 be AR,A,AR Given thatSum of A P
=126⇒a-d+a+a+d=126⇒3a=126⇒a=42Second term of A
P +Second Term of G P
=76⇒a+A=76⇒42+A=76⇒A=34First term of A
P +First Term of G P
=85⇒a-d+AR=85⇒42-d+34R=85⇒34R-d=43 ;iThird term of A
P +Third Term of G P
=84⇒a+d+AR=84⇒42+d+34R=84⇒34R+d=42 ;iiii+i⇒34R+d+34R-d=42+43⇒34R+1R=85⇒34R2+1R=85⇒34R2-85R+34=0Dividing each term by 17 we get,2R2-5R+2=0⇒2R2-4R-R+2=0⇒2RR-2-1R-2=0⇒2R-1R-2=0⇒R=12 or R=2When R=2 and A=34, we haveAR,A,AR=342,34,34×2=17,34,68When R=12 and A=34, we haveAR,A,AR=3412,34,34×12=68,34,17Hence we will get same numbers for both values of R, just the order is reversed
Put R=2 in i,342-d=43⇒d=17-43⇒d=-26We have a=42 and d=-26, hencea-d,a,a+d=42--26,42,42-26=68,42,16Put R=12 in i,3412-d=43⇒d=68-43⇒d=25We have a=42 and d=25, hencea-d,a,a+d=42-25,42,42+25=17,42,67Hence Numbers in A
P  are 68,42,16 and numbers in G P
 are 17,34,68 ORNumbers in A P
 are 17,42,67 and numbers in G P
 are 68,34,17

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Three numbers are in AP. Another 3 numbers are in GP. The sum of first term of the AP and the first term of the GP is 85, the sum of second term of AP and the second term of the GP is 76 and that of the 3rd term of AP and 3rd term of GP is 84. The sum of the AP is 126. Find each term of AP and GP

Three numbers are in AP. Another 3 numbers are in GP. The sum of first term of the AP and
the first term of the GP is 85, the sum of second term of AP and the second term of the GP is
76 and that of the 3rd term of AP and 3rd term of GP is 84. The sum of the AP is 126. Find each
term of AP and GP