Q63. Let three real numbers $a , b , c$ be in arithmetic progression and $a + 1 , b , c + 3$ be in geometric progression. If $a > 10$ and the arithmetic mean of $a , b$ and $c$ is 8 , then the cube of the geometric mean of $a , b$ and $c$ is
(1) 128
(2) 316
(3) 120
(4) 312

If $a + b + c = 1$ and $a < b < c , a , b , c \in R$ and $a ^ { \mathbf { 2 } } , 2 b ^ { \mathbf { 2 } } , c ^ { \mathbf { 2 } }$ are in G.P. and $a , b , c$ are in A.P. then find the value of $9 \left( a ^ { 2 } + b ^ { 2 } + c ^ { 2 } \right) =$ ?

The first three terms of an arithmetic progression are $p , q$ and $p ^ { 2 }$ respectively, where $p < 0$
The first three terms of a geometric progression are $p , p ^ { 2 }$ and $q$ respectively.
Find the sum of the first 10 terms of the arithmetic progression.
A $\frac { 23 } { 8 }$
B $\frac { 95 } { 8 }$
C $\frac { 115 } { 8 }$
D $\frac { 185 } { 8 }$

A geometric sequence $(b_n)$ with first two terms $b_{1} = \frac{4}{3}$ and $b_{2} = 2$ and an arithmetic sequence $(a_n)$ whose common difference equals the common ratio of this geometric sequence are given.
If $b_{7} = a_{11}$, what is $a_{1}$?
A) $\frac{1}{4}$ B) $\frac{1}{8}$ C) $\frac{3}{8}$ D) $\frac{3}{16}$ E) $\frac{5}{16}$